text:
\section{Introduction}
The theory of cluster algebras are invented by Fomin and Zelevinsky in \cite{FZ02} as a combinatorial and algebraic framework for studying total positivity for semisimple algebraic groups developed by Lusztig \cite{L94} and canonical bases of quantum groups introduced by Lusztig \cite{L90} and Kashiwara \cite{K91}. It has many applications including quiver representations, Teichm\"{u}ller theory, tropical geometry, integrable systems, and Poisson geometry.

Let $\mathfrak{g}$ be the simple Lie algebra and $U_q \widehat{\mathfrak{g}}$ the corresponding quantum affine algebras. In a remarkable paper \cite{HL10}, Hernandez and Leclerc make some connections between cluster algebras and finite dimensional representations of $U_q \widehat{\mathfrak{g}}$. They show that the Grothendieck ring $\rep( U_q \widehat{\mathfrak{g}} )$ of some subcategory of the category of all finite dimensional representations of $U_q \widehat{\mathfrak{g}}$ has a cluster algebra structure. In the paper \cite{HL13}, they apply the theory of cluster algebras to study the $q$-characters a family of $U_q \widehat{\mathfrak{g}}$-modules called Kirillov-Reshetkhin modules and they give a new algorithm to compute the $q$-characters of these modules.

The family of minimal affinizations of quantum groups $U_q \mathfrak{g}$ is an important family of simple modules of $U_q \widehat{\mathfrak{g}}$ which was introduced in \cite{C95}. The family of minimal affinizations contains the celebrated Kirillov-Reshetikhin modules. Minimal affinizations are studied intensively in recent years, see for example, \cite{CMY13}, \cite{CG11}, \cite{Her07}, \cite{LM13}, \cite{M10}, \cite{MP11}, \cite{MY12a}, \cite{MY12b}, \cite{MY14}, \cite{Nao13}.

The aim of this paper is to make a connection between cluster algebras and minimal affinizations of the quantum affine algebras $U_q \widehat{\mathfrak{g}}$ of type $G_2$. First we introduce a system of equations which we call the M-system of type $G_2$ and prove that the equations in the M-system of type $G_2$ is satisfied by the $q$-characters of minimal affinizations of type $G_2$.

The extended T-system of type $G_2$ studied in \cite{LM13} contains all minimal affinizations of type $G_2$ and some other types of modules. The M-system of type $G_2$ also contains all minimal affinizations of type $G_2$. But unlike the extended T-system of type $G_2$, the M-system of type $G_2$ contains only minimal affinizations of type $G_2$. Moreover, the M-system of type $G_2$ is much simpler than the extended T-system of type $G_2$.

We show that the equations in the M-system of type $G_2$ correspond to mutations in some cluster algebra $\mathscr{A}$. The cluster algebra $\mathscr{A}$ is the same as the cluster algebra of type $G_2$ introduced in \cite{HL13}. The minimal affinizations correspond to some cluster variables in $\mathscr{A}$.

We also show that the modules in the summands on the right hand side of each equation in the M-system is simple. In the last part of the paper, we study the dual M-system of type $G_2$.

The M-systems also exist for other Dynkin types of minimal affinizations. Since the method of proving that the $q$-characters of minimal affinizations satisfy the M-systems of other types are different from the method used in this paper and the M-systems of other types are much more complicated, we will publish them in other papers.

The paper is organized as follows. In Section 2, we give some background information about cluster algebras and representation theory of quantum affine algebras. In Section \ref{main results}, we describe our main results in this paper. In Section \ref{proof main1} and \ref{proof irreducible} we prove two main theorems given in Section \ref{main results}. In Section \ref{dual M system}, we study the dual M-system of type $G_2$.

\section{Background}
\subsection{Cluster algebras}
Cluster algebras are invented by Fomin and Zelevinsky in \cite{FZ02}. Let $\mathbb{Q}$ be the field of rational numbers and $\mathcal{F}=\mathbb{Q}(x_{1}, x_{2}, \cdots, x_{n})$ the field of rational functions. A seed in $\mathcal{F}$ is a pair $\Sigma=({\bf y}, Q)$, where ${\bf y} = (y_{1}, y_{2}, \cdots, y_{n})$ is a free generating set of $\mathcal{F}$, and $Q$ is a quiver with vertices labeled by $\{1, 2, \cdots, n\}$. Assume that $Q$ has neither loops nor $2$-cycles. For $k=1, 2, \cdots, n$, one defines a mutation $\mu_k$ by $\mu_k({\bf y}, Q) = ({\bf y}', Q')$. Here ${\bf y}' = (y_1', \ldots, y_n')$, $y_{i}'=y_{i}$, for $i\neq k$, and
\begin{equation}
y_{k}'=\frac{\prod_{i\rightarrow k} y_{i}+\prod_{k\rightarrow j} y_{j}}{y_{k}}, \label{exchange relation}
\end{equation}
where the first (resp. second) product in the right hand side is over all arrows of $Q$ with target (resp. source) $k$, and $Q'$ is obtained from $Q$ by
\begin{enumerate}
\item[(i)] adding a new arrow $i\rightarrow j$ for every existing pair of arrow $i\rightarrow k$ and $k\rightarrow j$;

\item[(ii)] reversing the orientation of every arrow with target or source equal to $k$;

\item[(iii)] erasing every pair of opposite arrows possible created by (i).
\end{enumerate}
The mutation class $\mathcal{C}(\Sigma)$ is the set of all seeds obtained from $\Sigma$ by a finite sequence of mutation $\mu_{k}$. If $\Sigma'=((y_{1}', y_{2}', \cdots, y_{n}'), Q')$ is a seed in $\mathcal{C}(\Sigma)$, then the subset $\{y_{1}', y_{2}', \cdots, y_{n}'\}$ is called a $cluster$, and its elements are called \textit{cluster variables}. The \textit{cluster algebra} $\mathscr{A}_{\Sigma}$ as the subring of $\mathcal{F}$ generated by all cluster variables. \textit{Cluster monomials} are monomials in the cluster variables supported on a single cluster.

In this paper, the initial seed in the cluster algebra we use is of the form $\Sigma=({\bf y}, Q)$, where ${\bf y}$ is an infinite set and $Q$ is an infinite quiver.

\begin{definition}[{Definition 3.1, \cite{GG14}}] \label{definition of cluster algebras of infinite rank}
Let $Q$ be a quiver without loops or $2$-cycles and with a countably infinite number of vertices labelled by all integers $i \in \mathbb{Z}$. Furthermore, for each vertex $i$ of $Q$ let the
number of arrows incident with $i$ be finite. Let ${\bf y} = \{y_i \mid i \in \mathbb{Z}\}$. An infinite initial seed is the pair $({\bf y}, Q)$. By finite sequences of mutation
at vertices of $Q$ and simultaneous mutation of the set ${\bf y}$ using the exchange relation (\ref{exchange relation}), one obtains a family of infinite seeds. The sets of variables in these seeds are called
the infinite clusters and their elements are called the cluster variables. The cluster algebra of
infinite rank of type $Q$ is the subalgebra of $\mathbb{Q}({\bf y})$ generated by the cluster variables.
\end{definition}

\subsection{The quantum affine algebra of type $G_2$}

In this paper, we take $\mathfrak{g}$ to be the complex simple Lie algebra of type $G_2$ and $\mathfrak{h}$ a Cartan subalgebra of $\mathfrak{g}$. Let $I=\{1, 2\}$.
We choose simple roots $\alpha_1, \alpha_2$ and scalar product $(\cdot, \cdot)$ such that
\begin{align*}
( \alpha_1, \alpha_1 ) = 6, \ ( \alpha_1, \alpha_2 )=-3, \ ( \alpha_2, \alpha_2 )=2.
\end{align*}
Therefore $\alpha_1$ is the long simple root and $\alpha_2$ is the short simple root. Let $\{\alpha_1^{\vee}, \alpha_2^{\vee}\}$ and $\{\omega_1, \omega_2\}$ be the sets of simple coroots and fundamental weights respectively. Let $C=(C_{ij})_{i, j\in I}$ denote the Cartan matrix, where $C_{ij}=\frac{2 ( \alpha_i, \alpha_j ) }{( \alpha_i, \alpha_i )}$. Let $r_1=3, r_2=1$, $D=\diag (r_1, r_2)$ and $B=DC$.
Then
\begin{align*}
C=
\left(
\begin{array}{cc}
2 & -1 \\
-3 & 2
\end{array}
\right), \quad
B=
\left(
\begin{array}{cc}
6 & -3 \\
-3 & 2
\end{array}
\right).
\end{align*}

Let $Q$ (resp. $Q^+$) and $P$ (resp. $P^+$) denote the $\mathbb{Z}$-span (resp. $\mathbb{Z}_{\geq 0}$-span) of the simple roots and fundamental weights respectively. Let $\leq$ be the partial order on $P$ in which $\lambda \leq \lambda'$ if and only if $\lambda' - \lambda \in Q^+$.

Let $\widehat{\mathfrak{g}}$ denote the untwisted affine algebra corresponding to $\mathfrak{g}$. Fix a $q\in \mathbb{C}^{\times}$, not a root of unity. Let $q_i=q^{r_i}, i=1, 2$. Let $\mathcal{P}$ the free abelian multiplicative group of monomials in infinitely many formal variables $(Y_{i, a})_{i\in I, a\in \mathbb{C}^{\times}}$.

The quantum affine algebra $U_q\widehat{\mathfrak{g}}$ in Drinfeld's new realization, see \cite{Dri88}, is generated by $x_{i, n}^{\pm}$ ($i\in I, n\in \mathbb{Z}$), $k_i^{\pm 1}$ $(i\in I)$, $h_{i, n}$ ($i\in I, n\in \mathbb{Z}\backslash \{0\}$) and central elements $c^{\pm 1/2}$, subject certain relations.

The quantum affine algebra $U_q\widehat{\mathfrak{g}}$ contains two standard quantum affine algebras of type $A_1$. The first one is $U_{q_1} \widehat{\mathfrak{sl}_2}$ generated by $x_{1, n}^{\pm}$ ($n\in \mathbb{Z}$), $k_1^{\pm 1}$, $h_{1, n}$ ($n\in \mathbb{Z}\backslash \{0\}$) and central elements $c^{\pm 1/2}$. The second one is $U_{q_2} \widehat{\mathfrak{sl}_2} $ generated by $x_{2, n}^{\pm}$ ($n\in \mathbb{Z}$), $k_2^{\pm 1}$, $h_{2, n}$ ($n\in \mathbb{Z}\backslash \{0\}$) and central elements $c^{\pm 1/2}$.

The subalgebra of $U_q\widehat{\mathfrak{g}}$ generated by $(k_i^{\pm})_{i\in I}, (x_{i, 0}^{\pm})_{i\in I}$ is a Hopf subalgebra of $U_q\widehat{\mathfrak{g}}$ and is isomorphic as a Hopf algebra to $U_q\mathfrak{g}$, the quantized enveloping algebra of $\mathfrak{g}$. In this way, $U_q\widehat{\mathfrak{g}}$-modules restrict to $U_q\mathfrak{g}$-modules.

\subsection{Finite-dimensional representations of $U_q \widehat{\mathfrak{g}}$ and $q$-characters} In this section, we recall the standard facts about finite-dimensional representations of $U_q\widehat{\mathfrak{g}}$ and $q$-characters of these representations, see \cite{CP94}, \cite{CP95a}, \cite{FR98}, \cite{MY12a}.

A representation $V$ of $U_q\widehat{\mathfrak{g}}$ is of type $1$ if $c^{\pm 1/2}$ acts as the identity on $V$ and
\begin{align} \label{decomposition}
V=\bigoplus_{\lambda \in P} V_{\lambda}, \ V_{\lambda} = \{v\in V  :   k_i v=q^{( \alpha_i, \lambda )} v \}.
\end{align}
In the following, all representations will be assumed to be finite-dimensional and of type $1$ without further comment. The decomposition (\ref{decomposition}) of a finite-dimensional representation $V$ into its $U_q\mathfrak{g}$-weight spaces can be refined by decomposing it into the Jordan subspaces of the mutually commuting operators $\phi_{i, \pm r}^{\pm}$, see \cite{FR98}:
\begin{align}
V=\bigoplus_{\gamma} V_{\gamma}, \ \gamma=(\gamma_{i, \pm r}^{\pm})_{i\in I, r\in \mathbb{Z}_{\geq 0}}, \ \gamma_{i, \pm r}^{\pm} \in \mathbb{C},
\end{align}
where
\begin{align*}
V_{\gamma} = \{v\in V  :   \exists k\in \mathbb{N}, \forall i \in I, m \geq 0, (\phi_{i, \pm m}^{\pm} - \gamma_{i, \pm m}^{\pm})^{k} v =0 \}.
\end{align*}
Here $\phi_{i, n}^{\pm}$'s are determined by the formula
\begin{align}
\phi_i^{\pm}(u) := \sum_{n=0}^{\infty} \phi_{i, \pm n}^{\pm} u^{\pm n} = k_i^{\pm 1} \exp\left( \pm (q-q^{-1}) \sum_{m=1}^{\infty} h_{i, \pm m}u^{\pm m}\right).
\end{align}
If $\dim(V_{\gamma})>0$, then $\gamma$ is called an \textit{$l$-weight} of $V$. For every finite dimensional representation of $U_q\widehat{\mathfrak{g}}$, the $l$-weights are known, see \cite{FR98}, to be of the form
\begin{align}
\gamma_i^{\pm}(u) := \sum_{r=0}^{\infty} \gamma_{i, \pm r}^{\pm} u^{\pm r} = q_i^{\deg Q_i - \deg R_i} \frac{Q_i(uq_i^{-1})R_i(uq_i)}{Q_{i}(uq_i) R_{i}(uq_i^{-1})}, \label{gamma}
\end{align}
where the right hand side is to be treated as a formal series in positive (resp. negative) integer powers of $u$, and $Q_i, R_i$ are polynomials of the form
\begin{align}
Q_i(u) = \prod_{a\in \mathbb{C}^{\times}} (1-ua)^{w_{i, a}}, \ R_{i}(u)=\prod_{a\in \mathbb{C}^{\times}}(1-ua)^{x_{i, a}}, \label{QR}
\end{align}
for some $w_{i, a}, x_{i, a} \in \mathbb{Z}_{\geq 0}, i\in I, a\in \mathbb{C}^{\times}$. Let $\mathcal{P}$ denote the free abelian multiplicative group of monomials in infinitely many formal variables $(Y_{i, a})_{i\in I, a\in \mathbb{C}^{\times}}$. There is a bijection $\gamma$ from $\mathcal{P}$ to the set of $l$-weights of finite-dimensional modules such that for the monomial $m=\prod_{i\in I, a\in \mathbb{C}^{\times}} Y_{i, a}^{w_{i, a}-x_{i, a}}$, the $l$-weight $\gamma(m)$ is given by (\ref{gamma}), (\ref{QR}).

Let $\mathbb{Z}\mathcal{P} = \mathbb{Z}[Y_{i, a}^{\pm 1}]_{i\in I, a\in \mathbb{C}^{\times}}$ be the group ring of $\mathcal{P}$. For $\chi \in \mathbb{Z}\mathcal{P}$, we write $m\in \mathcal{P}$ if the coefficient of $m$ in $\chi$ is non-zero.

The $q$-character of a $U_q\widehat{\mathfrak{g}}$-module $V$ is given by
\begin{align*}
\chi_q(V) = \sum_{m\in \mathcal{P}} \dim(V_{m}) m \in \mathbb{Z}\mathcal{P},
\end{align*}
where $V_m = V_{\gamma(m)}$.

Let $\rep(U_q\widehat{\mathfrak{g}})$ be the Grothendieck ring of finite-dimensional representations of $U_q\widehat{\mathfrak{g}}$ and $[V]\in \rep(U_q\widehat{\mathfrak{g}})$ the class of a finite-dimensional $U_q\widehat{\mathfrak{g}}$-module $V$. The $q$-character map defines an injective ring homomorphism, see \cite{FR98},
\begin{align*}
\chi_q: \rep(U_q\widehat{\mathfrak{g}}) \to \mathbb{Z}\mathcal{P}.
\end{align*}

For any finite-dimensional representation $V$ of $U_q\widehat{\mathfrak{g}}$, denote by $\mathscr{M}(V)$ the set of all monomials in $\chi_q(V)$. For each $j\in I$, a monomial $m=\prod_{i\in I, a\in \mathbb{C}^{\times}} Y_{i, a}^{u_{i, a}}$, where $u_{i, a}$ are some integers, is said to be \textit{$j$-dominant} (resp. \textit{$j$-anti-dominant}) if and only if $u_{j, a} \geq 0$ (resp. $u_{j, a} \leq 0$) for all $a\in \mathbb{C}^{\times}$. A monomial is called \textit{dominant} (resp. \textit{anti-dominant}) if and only if it is $j$-dominant (resp. $j$-anti-dominant) for all $j\in I$. Let $\mathcal{P}^+ \subset \mathcal{P}$ denote the set of all dominant monomials.

Let $V$ be a representation of $U_q\widehat{\mathfrak{g}}$ and $m\in \mathscr{M}(V)$ a monomial. A non-zero vector $v\in V_m$ is called a \textit{highest $l$-weight vector} with \textit{highest $l$-weight} $\gamma(m)$ if
\begin{align*}
x_{i, r}^{+} \cdot v=0, \ \phi_{i, \pm t}^{\pm} \cdot v=\gamma(m)_{i, \pm t}^{\pm} v, \ \forall i\in I, r\in \mathbb{Z}, t\in \mathbb{Z}_{\geq 0}.
\end{align*}
The module $V$ is called a \textit{highest $l$-weight representation} if $V=U_q\widehat{\mathfrak{g}}\cdot v$ for some highest $l$-weight vector $v\in V$.

It is known, see \cite{CP94}, \cite{CP95a}, that for each $m_+\in \mathcal{P}^{+}$ there is a unique finite-dimensional irreducible representation, denoted $L(m_+)$, of $U_q\widehat{\mathfrak{g}}$ that is highest $l$-weight with highest $l$-weight $\gamma(m_+)$, and moreover every finite-dimensional irreducible $U_q\widehat{\mathfrak{g}}$-module is of this form for some $m_+ \in \mathcal{P}^+$. Also, if $m_+, m_+' \in \mathcal{P}^{+}$ and $m_{+}\neq m_{+}'$, then $L(m_+) \not\cong L(m_+')$. For $m_+ \in \mathcal{P}^+$, we use $\chi_q(m_+)$ to denote $\chi_q(L(m_+))$. If $m \in \mathscr{M}(\chi_q(m_+))$, then we write $m \in \chi_q(m_+)$.

The following lemma is well-known.
\begin{lemma}
\label{contains in a larger set}
Let $m_1, m_2$ be two monomials. Then $L(m_1m_2)$ is a sub-quotient of $L(m_1) \otimes L(m_2)$. In particular, $\mathscr{M}(L(m_1m_2)) \subseteq \mathscr{M}(L(m_1))\mathscr{M}(L(m_2))$.   $\Box$
\end{lemma}

For $b\in \mathbb{C}^{\times}$, define the shift of spectral parameter map $\tau_b: \mathbb{Z}\mathcal{P} \to \mathbb{Z}\mathcal{P}$ to be a homomorphism of rings sending $Y_{i, a}^{\pm 1}$ to $Y_{i, ab}^{\pm 1}$. Let $m_1, m_2 \in \mathcal{P}^+$. If $\tau_b(m_1) = m_2$, then
\begin{align} \label{shift}
\tau_b \chi_q(m_1) = \chi_q(m_2).
\end{align}

A finite-dimensional $U_q\widehat{\mathfrak{g}}$-module $V$ is said to be \textit{special} if and only if $\mathscr{M}(V)$ contains exactly one dominant monomial. It is called \textit{anti-special} if and only if $\mathscr{M}(V)$ contains exactly one anti-dominant monomial. It is said to be \textit{prime} if and only if it is not isomorphic to a tensor product of two non-trivial $U_q\widehat{\mathfrak{g}}$-modules, see \cite{CP97}. Clearly, if a module is special or anti-special, then it is simple.

Define $A_{i, a} \in \mathcal{P}, i\in I, a\in \mathbb{C}^{\times}$, by
\begin{align*}
A_{1, a} = Y_{1, aq}Y_{1, aq^{-1}} Y_{2, a}^{-1}, \qquad
A_{2, a} = Y_{2, aq^{3}}Y_{2, aq^{-3}} Y_{1, aq^{-2}}^{-1} Y_{1, a}^{-1} Y_{1, aq^{2}}^{-1}.
\end{align*}
Let $\mathcal{Q}$ be the subgroup of $\mathcal{P}$ generated by $A_{i, a}, i\in I, a\in \mathbb{C}^{\times}$. Let $\mathcal{Q}^{\pm}$ be the monoids generated by $A_{i, a}^{\pm 1}, i\in I, a\in \mathbb{C}^{\times}$. There is a partial order $\leq$ on $\mathcal{P}$ in which
\begin{align}
m\leq m' \text{ if and only if } m'm^{-1}\in \mathcal{Q}^{+}. \label{partial order of monomials}
\end{align}
For all $m_+ \in \mathcal{P}^+$, $\mathscr{M}(L(m_+)) \subset m_+\mathcal{Q}^{-}$, see \cite{FM01}.

Let $m$ be a monomial. If for all $a \in \mathbb{C}^{\times}$ and $i\in I$, we have the property: if the power of $Y_{i, a}$ in $m$ is non-zero and the power of $Y_{j, aq^k}$ in $m$ is zero for all $j \in I, k\in \mathbb{Z}_{>0}$, then the power of $Y_{i, a}$ in $m$ is negative, then the monomial $m$ is called \textit{right negative}. For $i\in I, a\in \mathbb{C}^{\times}$, $A_{i,a}^{-1}$ is right-negative. A product of right-negative monomials is right-negative. If $m$ is right-negative and $m'\leq m$, then $m'$ is right-negative.

\subsection{Minimal affinizations of $U_q \mathfrak{g}$-modules} \label{definition of minimal affinizations}
Let $\lambda = k\omega_1 + l \omega_2$. A simple $U_q \widehat{\mathfrak{g}} $-module $L(m_+)$ is called a \textit{minimal affinization} of $V(\lambda)$ if and only if $m_+$ is one of the following monomials
\begin{align*}
\left( \prod_{i=0}^{k-1} Y_{1, aq^{6i}} \right) \left( \prod_{i=0}^{l-1} Y_{2, aq^{6k+2i+1}} \right), \qquad
\left( \prod_{i=0}^{l-1} Y_{2, aq^{2i}} \right) \left( \prod_{i=0}^{k-1} Y_{1, aq^{2l+6i+5}} \right),
\end{align*}
for some $a\in \mathbb{C}^{\times}$, see \cite{CP95b}.

From now on, we fix an $a\in \mathbb{C}^{\times}$ and denote $i_s = Y_{i, aq^s}$, $i \in I$, $s \in \mathbb{Z}$. Without loss of generality, we may assume that a simple $U_q \widehat{\mathfrak{g}} $-module $L(m_+)$ is a minimal affinization of $V(\lambda)$ if and only if $m_+$ is one of the following monomials
\begin{align*}
T_{k, l}^{(s)}=\left( \prod_{i=0}^{k-1}1_{s+6i} \right) \left( \prod_{j=0}^{l-1}2_{s+6k+2j+1} \right), \quad
\widetilde{T}_{k, l}^{(s)}=\left(\prod_{i=0}^{l-1}2_{-s-6k-2i-1}\right)\left(\prod_{j=0}^{k-1}1_{-s-6j}\right).
\end{align*}

\subsection{$q$-characters of $U_q \widehat{\mathfrak{sl}}_2$-modules and the Frenkel-Mukhin algorithm}
We recall the results of the $q$-characters of $U_q \widehat{\mathfrak{sl}}_2$-modules which are well-understood, see \cite{CP91}, \cite{FR98}.

Let $W_{k}^{(a)}$ be the irreducible representation $U_q \hat{\mathfrak{sl}}_2$ with
highest weight monomial
\begin{align*}
X_{k}^{(a)}=\prod_{i=0}^{k-1} Y_{aq^{k-2i-1}},
\end{align*}
where $Y_a=Y_{1, a}$. Then the $q$-character of $W_{k}^{(a)}$ is given by
\begin{align*}
\chi_q(W_{k}^{(a)})=X_{k}^{(a)} \sum_{i=0}^{k} \prod_{j=0}^{i-1} A_{aq^{k-2j}}^{-1},
\end{align*}
where $A_a=Y_{aq^{-1}}Y_{aq}$.

For $a\in \mathbb{C}^{\times}, k\in \mathbb{Z}_{\geq 1}$, the set $\Sigma_{k}^{(a)}=\{aq^{k-2i-1}\}_{i=0, \ldots, k-1}$ is called a \textit{string}. Two strings $\Sigma_{k}^{(a)}$ and $\Sigma_{k'}^{(a')}$ are said to be in \textit{general position} if the union $\Sigma_{k}^{(a)} \cup \Sigma_{k'}^{(a')}$ is not a string or $\Sigma_{k}^{(a)} \subset \Sigma_{k'}^{(a')}$ or $\Sigma_{k'}^{(a')} \subset \Sigma_{k}^{(a)}$.

Denote by $L(m_+)$ the irreducible $U_q \widehat{\mathfrak{sl}}_2$-module with
highest weight monomial $m_+$. Let $m_{+} \neq 1$ and $\in \mathbb{Z}[Y_a]_{a\in \mathbb{C}^{\times}}$ be a dominant monomial. Then $m_+$ can be uniquely (up to permutation) written in the form
\begin{align*}
m_+=\prod_{i=1}^{s} \left( \prod_{b\in \Sigma_{k_i}^{(a_i)}} Y_{b} \right),
\end{align*}
where $s$ is an integer, $\Sigma_{k_i}^{(a_i)}, i=1, \ldots, s$, are strings which are pairwise in general position and
\begin{align*}
L(m_+)=\bigotimes_{i=1}^s W_{k_i}^{(a_i)}, \qquad \chi_q(L(m_+))=\prod_{i=1}^s
 \chi_q(W_{k_i}^{(a_i)}).
\end{align*}

For $j\in I$, let
\begin{align*}
\beta_j : \mathbb{Z}[Y_{i, a}^{\pm 1}]_{i\in I; a\in \mathbb{C}^{\times}} \to \mathbb{Z}[Y_{a}^{\pm 1}]_{a\in \mathbb{C}^{\times}}
\end{align*}
be the ring homomorphism which sends, for all $a\in \mathbb{C}^{\times}$, $Y_{k, a} \mapsto 1$ for $k\neq j$ and $Y_{j, a} \mapsto Y_{a}$.

Let $V$ be a $U_q \widehat{\mathfrak{g}}$-module. Then $\beta_i(\chi_q(V))$, $i=1, 2$, is the $q$-character of $V$ considered as a $U_{q_i} \widehat{\mathfrak{sl}_2} $-module.

In some situation, we can use the $q$-characters of $U_q \widehat{\mathfrak{sl}}_2$-modules to compute the $q$-characters of $U_q \widehat{\mathfrak{g}} $-modules for arbitrary $\mathfrak{g}$,  see Section 5 in \cite{FM01}. The corresponding algorithm is called the Frenkel-Mukhin algorithm. The Frenkel-Mukhin algorithm recursively computes the minimal possible $q$-character which contains $m_+$ and is consistent when restricted to $U_{q_i} \widehat{\mathfrak{sl}_2}$, $i=1, 2$. For example, if a module $L(m_+)$ is special, then the Frenkel-Mukhin algorithm applied to $m_+$, see \cite{FM01}, produces the correct $q$-character $\chi_q(L(m_+))$.

\section{Main results} \label{main results}
In this section, we describe our main results.

\subsection{The $M$-system of type $G_2$}
We use $\mathcal T_{k, l}^{(s)}$ to denote the irreducible finite-dimensional $U_{q}\widehat{\mathfrak{g}}$-module with highest $l$-weight $T_{k, l}^{(s)}$. Here $T_{k, l}^{(s)}$ is defined in Section \ref{definition of minimal affinizations}. Let $[\mathcal{T}]$ be the equivalence class of the $U_q \widehat{\mathfrak{g}}$-module $\mathcal{T}$ in the Grothendieck ring $\rep( U_q \widehat{\mathfrak{g}} )$. Our first main result is the following system which we called $M$-system of type $G_2$.
\begin{theorem}\label{M system of type G2}
For $s\in \mathbb{Z}$, we have the following system of equations:
\begin{eqnarray}
[\mathcal T_{k, l}^{(s)}][\mathcal T_{k, 0}^{(s+6)}]&=&[\mathcal T_{k+1, 0}^{(s)}][\mathcal T_{k-1, l}^{(s+6)}]+[\mathcal T_{0, 3k+l}^{(s)}] \quad (k \in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\}), \label{eqn1}\\
{[\mathcal T_{k, l+3}^{(s)}]}[\mathcal T_{k, l}^{(s+6)}]&=&[\mathcal T_{k+1, l}^{(s)}][\mathcal T_{k-1, l+3}^{(s+6)}]+[\mathcal T_{0, l}^{(s+6k+6)}][\mathcal T_{0, 3k+l+3}^{(s)}] \quad (k, l \in \mathbb{Z}_{\geq 1}) \label{eqn2}.
\end{eqnarray}
\end{theorem}
Theorem \ref{M system of type G2} will be prove in Section \ref{proof main1}. The equations in Theorem \ref{M system of type G2} are equivalent to the following equations.
\begin{equation*}
\begin{split}
&\chi_q(\mathcal T_{k, l}^{(s)}) \chi_q(\mathcal T_{k, 0}^{(s+6)})=\chi_q(\mathcal T_{k+1, 0}^{(s)}) \chi_q(\mathcal T_{k-1, l}^{(s+6)})+\chi_q(\mathcal T_{0, 3k+l}^{(s)}) \quad (k \in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\}), \\
&\chi_q(\mathcal T_{k, l+3}^{(s)} ) \chi_q( \mathcal T_{k, l}^{(s+6)})=\chi_q(\mathcal T_{k+1, l}^{(s)}) \chi_q(\mathcal T_{k-1, l+3}^{(s+6)})+\chi_q(\mathcal T_{0, l}^{(s+6k+6)}) \chi_q(\mathcal T_{0, 3k+l+3}^{(s)})\,(k, l \in \mathbb{Z}_{\geq 1}).
\end{split}
\end{equation*}

\begin{example}\label{eample1}
The following are some equations in the $M$-system of type $G_2$.
\begin{eqnarray*}
\begin{split}
&[1_{-7}2_{0}][1_{-1}]=[1_{-7}1_{-1}][2_{0}]+[2_{-6}2_{-4}2_{-2}2_{0}],\\
&{[1_{-9}2_{-2}2_{0}]}[1_{-3}]=[1_{-9}1_{-3}][2_{-2}2_{0}]+[2_{-8}2_{-6}2_{-4}2_{-2}2_{0}],\\
&{[1_{-11}2_{-4}2_{-2}2_{0}]}[1_{-5}]=[1_{-11}1_{-5}][2_{-4}2_{-2}2_{0}]+[2_{-10}2_{-8}2_{-6}2_{-4}2_{-2}2_{0}],\\
&{[1_{-13}2_{-6}2_{-4}2_{-2}2_{0}]}[1_{-7}2_{0}]=[1_{-13}1_{-7}2_{0}][2_{-6}2_{-4}2_{-2}2_{0}]+[2_{0}][2_{-12}2_{-10}\cdots2_{-2}2_{0}],\\
&{[1_{-33}1_{-27}2_{-20}\cdots2_{-2}2_{0}]}[1_{-27}1_{-21}2_{-14}\cdots2_{-2}2_{0}]\\
&=[1_{-33}1_{-27}1_{-21}2_{-14}\cdots2_{-2}2_{0}][1_{-27}2_{-20}\cdots2_{-2}2_{0}]+[2_{-14}2_{-12}\cdots2_{-2}2_{0}][2_{-32}2_{-30}\cdots2_{-2}2_{0}].
\end{split}
\end{eqnarray*}
\end{example}

Moreover, we have the following theorem.
\begin{theorem}\label{irreducible}
For each relation in Theorem $\ref{M system of type G2}$, all summands on the right hand side are irreducible.
\end{theorem}
Theorem \ref{irreducible} will be prove in Section \ref{proof irreducible}.

\subsection{Definition of a cluster algebra $\mathscr{A}$} \label{definition of cluster algebra A}
Let $S=\{-2n+1\mid n\in \mathbb{Z}_{\geq 1}\}$, $S'=\{-2n+2 \mid n\in \mathbb{Z}_{\geq 1}\}$, and $V=(\{1\}\times S)\bigcup (\{2\}\times S')$. We define a quiver $Q$ with vertex set ${V}$ as follows. The arrows of $Q$ are given by the following rules. For $s_1, s_{2} \in S, s'_{1}, s'{_2} \in S'$, there is an arrow from $(1, s_1)$ to $(1, s_2)$ if and only if $s_2 = s_1 +6$ , there is an arrow from $(2, s'_1)$ to $(2, s'_2 )$ if and only if $s'_2 = s'_1 + 2 $, there is an arrow from $(1, s_1)$ to $(2, s'_1)$ if and only if $s'_1 = s_1 -5$, and there is an arrow from $(2, s'_2)$ to $(1, s_2 )$ if and only if $s_2 = s'_2 - 1 $. The quiver $Q$ is the same as the quiver $G^-$ of type $G_2$ in \cite{HL13}.

Let ${\bf t}=\{t_{k, 0}^{(s_1)}, t_{0, l}^{(s_2)} \mid s_1, s_2 \in S, k, l\in \mathbb{Z}_{\geq 1}\}$. Let $\mathscr{A}$ be the cluster algebra defined by the initial seed $({\bf t}, Q)$. By Definition \ref{definition of cluster algebras of infinite rank}, $\mathscr{A}$ is the $\mathbb{Q}$-subalgebra of the field of rational functions $\mathbb{Q}({\bf t})$ generated by all the elements obtained from some elements of $\bf t$ via a finite sequence of seed mutations. The cluster algebra $\mathscr{A}$ is the same as the cluster algebra of type $G_2$ introduced in \cite{HL13}.

\subsection{Mutation sequences}
We use $``C_{1}"$ to denote the column of vertices $(1, -1)$, $(1, -7)$, $\ldots$, $(1, -6n+5)$, $\cdots$ in the quiver $Q$. We use $``C_{2}"$ to denote the column of vertices $(1, -3)$, $(1, -9)$, $\ldots$, $(1, -6n+3)$, $\cdots$ in $Q$. We use $``C_{3}"$ to denote the column of vertices $(1, -5)$, $(1, -11)$, $\ldots$, $(1, -6n+1)$, $\cdots$ in $Q$. We use $``C_{4}"$ to denote the column of vertices $(2, 0)$, $(2, -2)$, $\ldots$, $(1, -2n+2)$, $\cdots$ in $Q$. By saying that mutate at the column $C_i$, $i \in \{1, 2, 3, 4\}$, we mean that we mutate the vertices of $C_i$ as follows. First we mutate at the first vertex in this column, then the second vertex, an so on until the vertex at infinity. By saying that we mutate $(C_{i_1}, C_{i_2}, \ldots, C_{i_n} )$, where $i_j \in \{1, 2, 3, 4\}, j=1,2,\ldots, n$, we mean that we first mutate the column $C_{i_1}$, then the column $C_{i_2}$, an so on up to the column $C_{i_n}$.

We define some variables $t_{k,l}^{(s)}$ ( $k, l \in \mathbb{Z}_{\geq 0}$, $s \in S$ ) recursively as follows. The variables $t_{k,0}^{(s_1)}$, $t_{0,l}^{(s_2)}$, $s_1, s_2 \in S$, are already defined. They are cluster variables in the initial seed of $\mathscr{A}$ define in Section \ref{definition of cluster algebra A}. For convenience, we write $t_{\lceil -s_1/6 \rceil, 0}^{(s_1)}$ at the vertex $(1,s_1)$ and write $t_{0, (-s_2+1)/2}^{(s_2)}$ at the vertex $(2,s_2)$ in the initial quiver $Q$, $s_1 , s_2 \in S$. Then we obtain the quiver (a) in Figure $\ref{figure}$.

Consider the mutation sequence $(C_{1}, C_{1}, \ldots,  C_{1})$ start from the initial seed, where the number of $C_1$ is $n$. First we mutate the first vertex in $C_1$ and define $t_{1, 1}^{(-7)} = {t'}_{1, 0}^{(-1)}$. Therefore
\begin{eqnarray}\label{eqn3}
t_{1, 1}^{(-7)} = {t'}_{1, 0}^{(-1)} = \frac{t_{2, 0}^{(-7)}t_{0, 1}^{(-1)}+t_{0, 4}^{(-7)}}{t_{1, 0}^{(-1)}}.
\end{eqnarray}
After this mutation, the quiver (a) in Figure $\ref{figure}$ becomes the quiver (b) in Figure $\ref{figure}$. Then we mutate the second vertex of $C_1$ and define $t_{2, 1}^{(-13)} = t'{_{2, 0}^{(-7)}}$. Therefore
\begin{eqnarray}\label{eqn4}
t_{2, 1}^{(-13)} = t'{_{2, 0}^{(-7)}} = \frac{t_{3, 0}^{(-13)}t_{1, 1}^{(-7)}+t_{0, 7}^{(-13)}}{t_{2, 0}^{(-7)}}.
\end{eqnarray}
After this mutation, the quiver (b) in Figure 1 becomes the quiver (c) in Figure $\ref{figure}$. We continue this procedure and mutate the vertices of $C_1$ in order and define $t_{k, 1}^{(-6k-1)} = t'{_{k, 0}^{(-6k+5)}}$ ($k=3,4,\ldots$) recursively. Therefore
\begin{eqnarray}\label{eqn5}
t_{k, 1}^{(-6k-1)} = t'{_{k, 0}^{(-6k+5)}} = \frac{t_{k+1, 0}^{(-6k-1)}t_{k-1, 1}^{(-6k+5)}+t_{0, 3k+1}^{(-6k-1)}}{t_{k, 0}^{(-6k+5)}}, \quad k = 3, 4, \ldots
\end{eqnarray}
Now we finish the mutation of the first $C_1$ in $(C_1, C_1, \ldots, C_1)$. We start to mutate the second $C_1$ in $(C_1, C_1, \ldots, C_1)$. First we mutate the first vertex in $C_1$ and define $t_{1, 4}^{(-13)} = {t'}_{1, 1}^{(-7)}$. Therefore
\begin{eqnarray}\label{eqn6}
t_{1, 4}^{(-13)} = {t'}_{1, 1}^{(-7)} = \frac{t_{2, 1}^{(-13)}t_{0, 4}^{(-7)}+t_{0, 1}^{(-1)}t_{0, 7}^{(-13)}}{t_{1, 1}^{(-7)}}.
\end{eqnarray}
After this mutation, we obtain the quiver (e) in Figure $\ref{figure}$. Then we mutate the second vertex of $C_1$ and define $t_{2, 4}^{(-19)} = {t'}_{2, 1}^{(-13)}$. Therefore
\begin{eqnarray}\label{eqn7}
t_{2, 4}^{(-19)} = {t'}_{2, 1}^{(-13)} = \frac{t_{3, 1}^{(-19)}t_{1, 4}^{(-13)}+t_{0, 1}^{(-1)}t_{0, 10}^{(-19)}}{t_{2, 1}^{(-13)}}.
\end{eqnarray}
After this mutation, the quiver (e) in Figure $\ref{figure}$ becomes the quiver (f) in Figure $\ref{figure}$. We continue this procedure and mutate vertices of $C_1$ in order and define $t_{k, 4}^{(-6k-7)} = t'{_{k, 1}^{(-6k-1)}}$ ($k=3,4,\ldots$) recursively. Therefore
\begin{eqnarray}\label{eqn8}
t_{k, 4}^{(-6k-7)} = t'{_{k, 1}^{(-6k-1)}} = \frac{t_{k+1, 1}^{(-6k-7)}t_{k-1, 4}^{(-6k-1)}+t_{0, 1}^{(-1)}t_{0, 3k+4}^{(-6k-7)}}{t_{k, 1}^{(-6k-1)}}, \quad k = 3,4, \ldots
\end{eqnarray}
Now we finish the mutation of the second $C_1$ in the mutation sequence $(C_1, C_1, \ldots, C_1)$. We continue this procedure and mutate $r$-th $(r=3, 4, \ldots, n)$ $C_1$ in $(C_1, C_1, \ldots, C_1)$ in order. We define $t_{k, 3r-2}^{(-6k-6r+5)} = t'{_{k, 3r-5}^{(-6k-6r+11)}}$, where $(k, r)=(1, 3), (2, 3), (3, 3), (4, 3), \ldots; (1, 4), (2, 4), (3, 4), (4, 4),\\ \ldots; (1, 5), (2, 5), (3, 5), (4, 5) \ldots; (1, 6), (2, 6), (3, 6), (4, 6), \ldots$, recursively. Therefore
\begin{eqnarray}\label{eqn9}
t_{k, 3r-2}^{(-6k-6r+5)} = t'{_{k, 3r-5}^{(-6k-6r+11)}} = \frac{t_{k+1, 3r-5}^{(-6k-6r+5)}t_{k-1, 3r-2}^{(-6k-6r+11)}+t_{0, 3r-5}^{(-6r+11)}t_{0, 3k+3r-2}^{(-6k-6r+5)}}{t{_{k, 3r-5}^{(-6k-6r+11)}}},
\end{eqnarray}
where $(k, r) = (1, 3), (2, 3), (3, 3), (4, 3), \ldots; (1, 4), (2, 4), (3, 4), (4, 4), \ldots; (1, 5), (2, 5), (3, 5), (4, 5)\\ \ldots; (1, 6), (2, 6), (3, 6), (4, 6), \ldots$

Similarly, we consider the mutation sequence $(C_{2}, C_{2}, \ldots, C_{2})$ start from the initial seed, where the number of $C_2$ is $n$. First we mutate vertices in the first $C_2$ in order and define $t_{k, 2}^{(-6k-3)} = t'{_{k, 0}^{(-6k+3)}}$ ($k = 1, 2, \ldots$) recursively. Therefore
\begin{eqnarray}\label{eqn10}
t_{k, 2}^{(-6k-3)} = t'{_{k, 0}^{(-6k+3)}}= \frac{t_{k+1, 0}^{(-6k-3)}t_{k-1, 2}^{(-6k+3)}+t_{0, 3k+2}^{(-6k-3)}}{t{_{k, 0}^{(-6k+3)}}}, \quad k = 1, 2, \ldots
\end{eqnarray}
Then we mutate the second $C_2$, the third $C_2$ and so on in the mutation sequence $(C_{2}, C_{2}, \ldots, C_{2})$. We define $t_{k, 3r-1}^{(-6k-6r+3)} = t'{_{k, 3r-4}^{(-6k-6r+9)}}$, where $(k, r)=(1, 2), (2, 2), (3, 2), (4, 2), \ldots; (1, 3), (2, 3),\\(3, 3), (4, 3), \ldots; (1, 4), (2, 4), (3, 4), (4, 4) \ldots; (1, 5), (2, 5), (3, 5), (4, 5), \ldots$, recursively. Therefore
\begin{eqnarray}\label{eqn11}
t_{k, 3r-1}^{(-6k-6r+3)} = t'{_{k, 3r-4}^{(-6k-6r+9)}}= \frac{t_{k+1, 3r-4}^{(-6k-6r+3))}t_{k-1, 3r-1}^{(-6k-6r+9)}+t_{0, 3r-4}^{(-6r+9)}t_{0, 3k+3r-1}^{(-6k-6r+3)}}{t{_{k, 3r-4}^{(-6k-6r+9)}}},
\end{eqnarray}
where $(k, r) = (1, 2), (2, 2), (3, 2), (4, 2), \ldots; (1, 3), (2, 3), (3, 3), (4, 3), \ldots; (1, 4), (2, 4), (3, 4), (4, 4)\\ \ldots; (1, 5), (2, 5), (3, 5), (4, 5), \ldots$

We consider the mutation sequence $(C_{3}, C_{3}, \ldots,  C_{3})$ start from the initial seed, where the number of $C_3$ is $n$. First we mutate vertices in the first $C_3$ in order and define $t_{k, 2}^{(-6k-5)} = t'{_{k, 0}^{(-6k+1)}}$ ($k = 1, 2, \ldots$) recursively. Therefore
\begin{eqnarray}\label{eqn12}
t_{k, 3}^{(-6k-5)} = t'{_{k, 0}^{(-6k+1)}}= \frac{t_{k+1, 0}^{(-6k-5)}t_{k-1, 3}^{(-6k+1)}+t_{0, 3k+3}^{(-6k-5)}}{t{_{k, 0}^{(-6k+1)}}}, \quad k = 1, 2, \ldots
\end{eqnarray}
Then we mutate the second $C_3$, the third $C_3$ and so on in the mutation sequence $(C_{3}, C_{3}, \cdots, C_{3})$. We define $t_{k, 3r}^{(-6k-6r+1)} = t'{_{k, 3r-3}^{(-6k-6r+7)}}$, where $(k, r) = (1, 2), (2, 2), (3, 2), (4, 2), \ldots; (1, 3), (2, 3),\\ (3, 3), (4, 3), \ldots; (1, 4), (2, 4), (3, 4), (4, 4), \ldots; (1, 5), (2, 5), (3, 5), (4, 5) \ldots$, recursively. Therefore
\begin{eqnarray}\label{eqn13}
t_{k, 3r}^{(-6k-6r+1)} = t'{_{k, 3r-3}^{(-6k-6r+7)}}= \frac{t_{k+1, 3r-3}^{(-6k-6r+1)}t_{k-1, 3r}^{(-6k-6r+7)}+t_{0, 3r-3}^{(-6r+7)}t_{0, 3k+3r}^{(-6k-6r+1)}}{t{_{k, 3r-3}^{(-6k-6r+7)}}},
\end{eqnarray}
where $(k, r) = (1, 2), (2, 2), (3, 2), (4, 2), \ldots; (1, 3), (2, 3), (3, 3), (4, 3), \ldots; (1, 4), (2, 4), (3, 4),\\ (4, 4), \ldots; (1, 5), (2, 5), (3, 5), (4, 5) \ldots$

\begin{figure}
\begin{minipage}[t]{.25\linewidth}
\begin{xy}
(13,90) *+{t_{0, 1}^{(-1)}} ="4";%
(13,80) *+{t_{0, 2}^{(-3)}} ="5";(26,80) *+{t_{1, 0}^{(-1)}} ="13";%
(0,70) *+{t_{1, 0}^{(-3)}} ="1";(13,70) *+{t_{0, 3}^{(-5)}} ="6";%
(13,60) *+{t_{0, 4}^{(-7)}} ="7";(39,60) *+{t_{1, 0}^{(-5)}} ="16";%
(13,50) *+{t_{0, 5}^{(-9)}} ="8";(26,50) *+{t_{2, 0}^{(-7)}} ="14";%
(0,40) *+{t_{2, 0}^{(-9)}} ="2";(13,40) *+{t_{0, 6}^{(-11)}} ="9";%
(13,30) *+{t_{0, 7}^{(-13)}} ="10";(39,30) *+{t_{2, 0}^{(-11)}} ="17";%
(13,20) *+{t_{0, 8}^{(-15)}} ="11";(26,20) *+{t_{3, 0}^{(-13)}} ="15";%
(0,10) *+{t_{3, 0}^{(-15)}} ="3";(13,10) *+{t_{0, 9}^{(-17)}} ="12";%
(0,0) *+{\vdots} ="18";(13,0) *+{\vdots} ="19";(20,-5) *+{(\text{a})} ="22";(26,0) *+{\vdots} ="20";(39,0) *+{\vdots} ="21"
{\ar "5";"4"};{\ar "6";"5"};{\ar "7";"6"};{\ar "8";"7"};{\ar "9";"8"};{\ar "10";"9"};{\ar "11";"10"};{\ar "12";"11"};{\ar "19";"12"};%
{\ar "2";"1"};{\ar "3";"2"};{\ar "18";"3"};%
{\ar "14";"13"};{\ar "15";"14"};{\ar "20";"15"};%
{\ar "17";"16"};{\ar "21";"17"};%
{\ar "5";"1"};{\ar "1";"8"};{\ar "8";"2"};{\ar "2";"11"};%
{\ar "4";"13"};{\ar "13";"7"};{\ar "7";"14"};{\ar "14";"10"};{\ar "10";"15"};%
{\ar "6";"16"};{\ar "16";"9"};{\ar "9";"17"};{\ar "17";"12"}
\end{xy}
\end{minipage}
\begin{minipage}[t]{.25\linewidth}
\begin{xy}
(13,90) *+{t_{0, 1}^{(-1)}} ="4";%
(13,80) *+{t_{0, 2}^{(-3)}} ="5";(26,80) *+{\fbox {$t_{1, 1}^{(-7)}$}}="13";%
(0,70) *+{t_{1, 0}^{(-3)}} ="1";(13,70) *+{t_{0, 3}^{(-5)}} ="6";%
(13,60) *+{t_{0, 4}^{(-7)}} ="7";(39,60) *+{t_{1, 0}^{(-5)}} ="16";%
(13,50) *+{t_{0, 5}^{(-9)}} ="8";(26,50) *+{t_{2, 0}^{(-7)}} ="14";%
(0,40) *+{t_{2, 0}^{(-9)}} ="2";(13,40) *+{t_{0, 6}^{(-11)}} ="9";%
(13,30) *+{t_{0, 7}^{(-13)}} ="10";(39,30) *+{t_{2, 0}^{(-11)}} ="17";%
(13,20) *+{t_{0, 8}^{(-15)}} ="11";(26,20) *+{t_{3, 0}^{(-13)}} ="15";%
(0,10) *+{t_{3, 0}^{(-15)}} ="3";(13,10) *+{t_{0, 9}^{(-17)}} ="12";%
(0,0) *+{\vdots} ="18";(13,0) *+{\vdots} ="19";(20,-5) *+{(\text{b})} ="22";(26,0) *+{\vdots} ="20";(39,0) *+{\vdots} ="21"
{\ar "5";"4"};{\ar "6";"5"};{\ar "7";"6"};{\ar "8";"7"};{\ar "9";"8"};{\ar "10";"9"};{\ar "11";"10"};{\ar "12";"11"};{\ar "19";"12"};%
{\ar "2";"1"};{\ar "3";"2"};{\ar "18";"3"};%
{\ar "13";"14"};{\ar "15";"14"};{\ar "20";"15"};%
{\ar "17";"16"};{\ar "21";"17"};%
{\ar "5";"1"};{\ar "1";"8"};{\ar "8";"2"};{\ar "2";"11"};%
{\ar "13";"4"};{\ar "7";"13"};{\ar "14";"10"};{\ar "10";"15"};%
{\ar "6";"16"};{\ar "16";"9"};{\ar "9";"17"};{\ar "17";"12"};%
{\ar@/_2pc/"4";"7"}
\end{xy}
\end{minipage}
\begin{minipage}[t]{.25\linewidth}
\begin{xy}
(13,90) *+{t_{0, 1}^{(-1)}} ="4";%
(13,80) *+{t_{0, 2}^{(-3)}} ="5";(26,80) *+{t_{1, 1}^{(-7)}} ="13";%
(0,70) *+{t_{1, 0}^{(-3)}} ="1";(13,70) *+{t_{0, 3}^{(-5)}} ="6";%
(13,60) *+{t_{0, 4}^{(-7)}} ="7";(39,60) *+{t_{1, 0}^{(-5)}} ="16";%
(13,50) *+{t_{0, 5}^{(-9)}} ="8";(26,50) *+{\fbox {$t_{2, 1}^{(-13)}$}} ="14";%
(0,40) *+{t_{2, 0}^{(-9)}} ="2";(13,40) *+{t_{0, 6}^{(-11)}} ="9";%
(13,30) *+{t_{0, 7}^{(-13)}} ="10";(39,30) *+{t_{2, 0}^{(-11)}} ="17";%
(13,20) *+{t_{0, 8}^{(-15)}} ="11";(26,20) *+{t_{3, 0}^{(-13)}} ="15";%
(0,10) *+{t_{3, 0}^{(-15)}} ="3";(13,10) *+{t_{0, 9}^{(-17)}} ="12";%
(0,0) *+{\vdots} ="18";(13,0) *+{\vdots} ="19";(20,-5) *+{(\text{c})} ="22";(26,0) *+{\vdots} ="20";(39,0) *+{\vdots} ="21"
{\ar "5";"4"};{\ar "6";"5"};{\ar "7";"6"};{\ar "8";"7"};{\ar "9";"8"};{\ar "10";"9"};{\ar "11";"10"};{\ar "12";"11"};{\ar "19";"12"};%
{\ar "2";"1"};{\ar "3";"2"};{\ar "18";"3"};%
{\ar "14";"13"};{\ar "14";"15"};{\ar "20";"15"};%
{\ar "17";"16"};{\ar "21";"17"};%
{\ar "5";"1"};{\ar "1";"8"};{\ar "8";"2"};{\ar "2";"11"};%
{\ar "13";"4"};{\ar "13";"10"};{\ar "7";"13"};{\ar "10";"14"};%
{\ar "6";"16"};{\ar "16";"9"};{\ar "9";"17"};{\ar "17";"12"};%
{\ar@/_2pc/"4";"7"}
\end{xy}
\end{minipage}

\begin{minipage}[t]{.10\linewidth}
\begin{xy}
(11,90) *+{t_{0, 1}^{(-1)}} ="4";%
(11,80) *+{t_{0, 2}^{(-3)}} ="5";(22,80) *+{t_{1, 1}^{(-7)}} ="13";%
(0,70) *+{t_{1, 0}^{(-3)}} ="1";(11,70) *+{t_{0, 3}^{(-5)}} ="6";%
(11,60) *+{t_{0, 4}^{(-7)}} ="7";(33,60) *+{t_{1, 0}^{(-5)}} ="16";%
(11,50) *+{t_{0, 5}^{(-9)}} ="8";(22,50) *+{t_{2, 1}^{(-13)}} ="14";%
(0,40) *+{t_{2, 0}^{(-9)}} ="2";(11,40) *+{t_{0, 6}^{(-11)}} ="9";%
(11,30) *+{t_{0, 7}^{(-13)}} ="10";(33,30) *+{t_{2, 0}^{(-11)}} ="17";%
(11,20) *+{t_{0, 8}^{(-15)}} ="11";(22,20) *+{\fbox {$t_{3, 1}^{(-19)}$}} ="15";%
(0,10) *+{t_{3, 0}^{(-15)}} ="3";(11,10) *+{t_{0, 9}^{(-17)}} ="12";%
(0,0) *+{\vdots} ="18";(11,0) *+{\vdots} ="19";(15,-5) *+{(\text{d})} ="22";(22,0) *+{\vdots} ="20";(33,0) *+{\vdots} ="21"
{\ar "5";"4"};{\ar "6";"5"};{\ar "7";"6"};{\ar "8";"7"};{\ar "9";"8"};{\ar "10";"9"};{\ar "11";"10"};{\ar "12";"11"};{\ar "19";"12"};%
{\ar "2";"1"};{\ar "3";"2"};{\ar "18";"3"};%
{\ar "14";"13"};{\ar "14";"15"};{\ar "20";"15"};%
{\ar "17";"16"};{\ar "21";"17"};%
{\ar "5";"1"};{\ar "1";"8"};{\ar "8";"2"};{\ar "2";"11"};%
{\ar "13";"4"};{\ar "13";"10"};{\ar "7";"13"};{\ar "10";"14"};%
{\ar "6";"16"};{\ar "16";"9"};{\ar "9";"17"};{\ar "17";"12"};%
{\ar@/_2pc/"4";"7"}
\end{xy}
\end{minipage}
\begin{minipage}[t]{.10\linewidth}
\begin{xy}
(2,60) *+{\cdots} ="4"
\end{xy}
\end{minipage}
\begin{minipage}[t]{.10\linewidth}
\begin{xy}
(11,92) *+{t_{0, 1}^{(-1)}} ="4";%
(11,80) *+{t_{0, 2}^{(-3)}} ="5";(22,80) *+{\fbox {$t_{1, 4}^{(-13)}$}} ="13";%
(0,70) *+{t_{1, 0}^{(-3)}} ="1";(11,70) *+{t_{0, 3}^{(-5)}} ="6";%
(11,60) *+{t_{0, 4}^{(-7)}} ="7";(33,60) *+{t_{1, 0}^{(-5)}} ="16";%
(11,50) *+{t_{0, 5}^{(-9)}} ="8";(22,50) *+{t_{2, 1}^{(-13)}} ="14";%
(0,40) *+{t_{2, 0}^{(-9)}} ="2";(11,40) *+{t_{0, 6}^{(-11)}} ="9";%
(11,30) *+{t_{0, 7}^{(-13)}} ="10";(33,30) *+{t_{2, 0}^{(-11)}} ="17";%
(11,20) *+{t_{0, 8}^{(-15)}} ="11";(22,20) *+{t_{3, 1}^{(-19)}} ="15";%
(0,10) *+{t_{3, 0}^{(-15)}} ="3";(11,10) *+{t_{0, 9}^{(-17)}} ="12";%
(0,0) *+{\vdots} ="18";(11,0) *+{\vdots} ="19";(15,-5) *+{(\text{e})} ="22";(22,0) *+{\vdots} ="20";(33,0) *+{\vdots} ="21"
{\ar "5";"4"};{\ar "6";"5"};{\ar "7";"6"};{\ar "8";"7"};{\ar "9";"8"};{\ar "10";"9"};{\ar "11";"10"};{\ar "12";"11"};{\ar "19";"12"};%
{\ar "2";"1"};{\ar "3";"2"};{\ar "18";"3"};%
{\ar "13";"14"};{\ar "14";"15"};{\ar "20";"15"};%
{\ar "17";"16"};{\ar "21";"17"};%
{\ar "5";"1"};{\ar "1";"8"};{\ar "8";"2"};{\ar "2";"11"};%
{\ar "4";"13"};{\ar "10";"13"};{\ar "13";"7"};{\ar "10";"14"};%
{\ar "6";"16"};{\ar "16";"9"};{\ar "9";"17"};{\ar "17";"12"};%
{\ar@/_2pc/"7";"10"}
\end{xy}
\end{minipage}
\begin{minipage}[t]{.10\linewidth}
\begin{xy}
(11,90) *+{t_{0, 1}^{(-1)}} ="4";%
(11,80) *+{t_{0, 2}^{(-3)}} ="5";(22,80) *+{t_{1, 4}^{(-13)}} ="13";%
(0,70) *+{t_{1, 0}^{(-3)}} ="1";(11,70) *+{t_{0, 3}^{(-5)}} ="6";%
(11,60) *+{t_{0, 4}^{(-7)}} ="7";(33,60) *+{t_{1, 0}^{(-5)}} ="16";%
(11,50) *+{t_{0, 5}^{(-9)}} ="8";(22,50) *+{\fbox {$t_{2, 4}^{(-19)}$}} ="14";%
(0,40) *+{t_{2, 0}^{(-9)}} ="2";(11,40) *+{t_{0, 6}^{(-11)}} ="9";%
(11,30) *+{t_{0, 7}^{(-13)}} ="10";(33,30) *+{t_{2, 0}^{(-11)}} ="17";%
(11,20) *+{t_{0, 8}^{(-15)}} ="11";(22,20) *+{t_{3, 1}^{(-19)}} ="15";%
(0,10) *+{t_{3, 0}^{(-15)}} ="3";(11,10) *+{t_{0, 9}^{(-17)}} ="12";%
(0,0) *+{\vdots} ="18";(11,0) *+{\vdots} ="19";(15,-5) *+{(\text{f})} ="22";(22,0) *+{\vdots} ="20";(33,0) *+{\vdots} ="21"
{\ar "5";"4"};{\ar "6";"5"};{\ar "7";"6"};{\ar "8";"7"};{\ar "9";"8"};{\ar "10";"9"};{\ar "11";"10"};{\ar "12";"11"};{\ar "19";"12"};%
{\ar "2";"1"};{\ar "3";"2"};{\ar "18";"3"};%
{\ar "13";"14"};{\ar "14";"15"};{\ar "20";"15"};%
{\ar "17";"16"};{\ar "21";"17"};%
{\ar "5";"1"};{\ar "1";"8"};{\ar "8";"2"};{\ar "2";"11"};%
{\ar "4";"13"};{\ar "10";"13"};{\ar "13";"7"};{\ar "10";"14"};%
{\ar "6";"16"};{\ar "16";"9"};{\ar "9";"17"};{\ar "17";"12"};%
{\ar@/_2pc/"7";"10"}
\end{xy}
\end{minipage}
\begin{minipage}[t]{.10\linewidth}
\begin{xy}
(2,60) *+{\cdots} ="4"
\end{xy}
\end{minipage}
\caption{The mutation sequence $(C_1, C_1, \ldots, C_1)$.}\label{figure}
\end{figure}

\subsection{The equations in the $M$-system of type $G_2$ correspond to mutations in the cluster algebra $\mathscr{A}$}

By (\ref{eqn3}), (\ref{eqn4}), (\ref{eqn5}), (\ref{eqn10}), (\ref{eqn12}), we have
\begin{eqnarray}
t_{k, l}^{(s)}=t'{_{k, 0}^{(s+6)}}=\frac{t_{k+1, 0}^{(s)}t_{k-1, l}^{(s+6)} + t_{0, 3k+l}^{(s)}}{t_{k, 0}^{(s+6)}}\qquad (k \in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\}), \label{eqn-def1}
\end{eqnarray}
where $s \in \{-2n-5 \mid n \in \mathbb{Z}_{\geq 1}\}$. Equations (\ref{eqn-def1}) correspond to Equations (\ref{eqn1}) in the M-system.

By (\ref{eqn6}), (\ref{eqn7}), (\ref{eqn8}), (\ref{eqn9}), (\ref{eqn11}), (\ref{eqn13}), we have
\begin{eqnarray}
t_{k, l+3}^{(s)}=t'{_{k, l}^{(s+6)}}=\frac{t_{k+1, l}^{(s)} t_{k-1, l+3}^{(s+6)}+t_{0, l}^{(s+6k+6)} t_{0, 3k+l+3}^{(s)}}{t_{k, l}^{(s+6)}}\qquad (k, l \in \mathbb{Z}_{\geq 1})\label{eqn-def2},
\end{eqnarray}
where $s \in \{-2n-11 \mid n \in \mathbb{Z}_{\geq 1}\}$. Equations (\ref{eqn-def2}) correspond to Equations (\ref{eqn2}) in the M-system. Therefore we have the following theorem.
\begin{theorem}
Minimal affinizations of type $G_2$ correspond to cluster variables in $\mathscr{A}$ defined in Section \ref{definition of cluster algebra A}.
\end{theorem}

\subsection{The $m$-system of type $G_2$}
For $k, l \in \mathbb{Z}_{\geq 0}$, let $m_{k,l} = \res(\mathcal{T}_{k,l}^{(0)})$ be the restriction of $\mathcal{T}_{k,l}^{(0)}$ to $U_q \mathfrak{g}$. Let $\chi(M)$ be the character of a $U_q \mathfrak{g}$-module $M$. By Theorem \ref{M system of type G2}, we have the following theorem.
\begin{theorem}\label{m system}
We have
\begin{eqnarray}
\chi(m_{k, l}) \chi( m_{k, 0}) &=& \chi(m_{k+1, 0}) \chi( m_{k-1, l}) + \chi( m_{0, 3k+l}) \quad (k \in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\}), \\
\chi( m_{k, l+3} ) \chi( m_{k, l} ) &=&  \chi( m_{k+1, l} ) \chi( m_{k-1, l+3} ) + \chi( m_{0, l} ) \chi( m_{0, 3k+l+3} ) \quad (k, l \in \mathbb{Z}_{\geq 1}).
\end{eqnarray}
\end{theorem}
We call the above system of equations the $m$-system of type $G_2$.
\section{Proof of Theorem $\ref{M system of type G2}$} \label{proof main1}
In this section, we prove Theorem $\ref{M system of type G2}$.

By the Frenkel-Mukhin algorithm, we have the following result.
\begin{lemma}\label{lemma}
The fundamental q-characters for $U_{q}\widehat{\mathfrak{g}}$ of type $G_{2}$ are given by
\begin{eqnarray*}
\chi_{q}(1_{0})&=&1_{0}+2_{1}2_{3}2_{5}1_{6}^{-1}+2_{1}2_{3}2_{7}^{-1}+2_{1}2_{5}^{-1}2_{7}^{-1}1_{4}+2_{3}^{-1}2_{5}^{-1}2_{7}^{-1}
1_{2}1_{4}\\
&&+2_{1}2_{9}1_{10}^{-1}+1_{4}1_{8}^{-1}+2_{3}^{-1}2_{9}1_{2}1_{10}^{-1}+2_{5}2_{7}2_{9}1_{8}^{-1}1_{10}^{-1}+2_{1}2_{11}^{-1}\\
&&+2_{3}^{-1}2_{11}^{-1}1_{2}+2_{5}2_{7}2_{11}^{-1}1_{8}^{-1}+2_{5}2_{9}^{-1}2_{11}^{-1}+2_{7}^{-1}2_{9}^{-1}2_{11}^{-1}1_{6}+1_{12}^{-1},\\
\chi_{q}(2_{0})&=&2_{0}+2_{2}^{-1}1_{1}+2_{4}2_{6}1_{7}^{-1}+2_{4}2_{8}^{-1}+2_{6}^{-1}2_{8}^{-1}1_{5}+2_{10}1_{11}^{-1}+2_{12}^{-1}.
\end{eqnarray*}
\end{lemma}

\subsection{Classification of dominant monomials in the summands on both sides of the M-system}By {Theorem 3.8 in \cite{Her07}} (see also {Theorem 3.3 in \cite{LM13}}), the modules $\mathcal T{_{k, l}^{(s)}}$ ($s\in \mathbb{Z}, k, l \in \mathbb{Z}_{\geq 0}$) are special. Therefore we can use the Frenkel-Mukhin algorithm to compute the $q$-characters of $\mathcal T{_{k, l}^{(s)}}$ ($s\in \mathbb{Z}, k, l \in \mathbb{Z}_{\geq 0}$).
Now we use the Frenkel-Mukhin algorithm to classify dominant monomials in the summands on both sides of the M-system.
\begin{lemma}\label{lemma1}
We have the following cases.
\begin{enumerate}[(1)]
\item Let $M =T_{k, l}^{(s)} T_{k, 0}^{(s+6)}$ $(k\in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\})$. Then dominant monomials in $\chi_q(T_{k, l}^{(s)}) \chi_q( T_{k, 0}^{(s+6)} )$ $(k\in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\})$ are $M$ and
\begin{equation*}
\begin{split}
&M_{r}=M\prod _{i=0}^{r-1}A_{1, aq^{-2l-6i-2}}^{-1}, \quad r=1, 2, \ldots, k.
\end{split}
\end{equation*}
The dominant monomials in $\chi_q(T_{k-1, l}^{(s+6)}) \chi_q(T_{k+1, 0}^{(s)})$ $(k\in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\})$ are $M$, $M_{1}$, $\ldots$, $M_{k-1}$. The only dominant monomial in $T_{0, 3k+l}^{(s)}$ $(k\in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\})$ is $M_{k}$.

\item Let $M=T_{k, l+3}^{(s)} T_{k, l}^{(s+6)}$ $(k, l \in \mathbb{Z}_{\geq 1})$. Then dominant monomials in $\chi_q(T_{k, l+3}^{(s)})\chi_q(T _{k, l}^{(s+6)})$ $(k, l \in \mathbb{Z}_{\geq 1})$ are $M$ and
\begin{equation*}
\begin{split}
M_{r}=M\prod _{i=0}^{r-1}A_{1, aq^{s+6k-6i-3}}^{-1}, \quad r=1, 2, \ldots, k.
\end{split}
\end{equation*}
The dominant monomials in $\chi_q(T_{k-1, l+3}^{(s+6)})\chi_q(T_{k+1, l}^{(s)})$ $(k, l \in \mathbb{Z}_{\geq 1})$ are $M$, $M_{1}$, $\ldots$, $M_{k-1}$. The only dominant monomial in $\chi_q(T_{0, l}^{(s+6k+6)})\chi_q(T_{0, 3k+l+3}^{(s)})$ $(k, l \in \mathbb{Z}_{\geq 1})$ is $M_{k}$.
\end{enumerate}
\end{lemma}

\begin{proof}
We prove the case of $\chi_{q}(\mathcal T_{k, l+3}^{(s)})\chi_{q}(\mathcal T_{k, l}^{(s+6)})$. The other cases are similar. Let $m_{1}'=T_{k, l+3}^{(s)}$, $m_{2}'=T_{k, l}^{(s+6)}$. Without loss of generality, we may assume that $s=0$. Then
\begin{equation*}
\begin{split}
&m_{1}'=(1_{0}1_{6}\cdots1_{6k-6})(2_{6k+1}2_{6k+3}\cdots2_{6k+2l+5}),\\
&m_{2}'=(1_{6}1_{12}\cdots1_{6k})(2_{6k+7}2_{6k+9}\cdots2_{6k+2l+5}).
\end{split}
\end{equation*}

Let $m=m_{1}m_{2}$ be a dominant monomial, where $m_{i}\in \chi_{q}(m_{i}'), i=1, 2$. We denote $m_{3}=2_{6k+1}2_{6k+7}\cdots 2_{6k+2l+5}$, $m_{4}=2_{6k+7}2_{6k+9}\cdots 2_{6k+2l+5}$.

Suppose that $m_{1}\in \chi_{q}(1_{0}1_{6}\cdots1_{6k-6})(\chi_{q}(m_{3})-m_{3})$, then $m=m_{1}m_{2}$ is right negative and hence $m$ is not dominant. This contradicts our assumption. Therefor $m_{1}\in \chi_{q}(1_{0}1_{6}\cdots1_{6k-6})m_{3}$. Similarly, if $m_{2}\in \chi_{q}(1_{6}1_{12}\cdots1_{6k})(\chi_{q}(m_{4})-m_{4})$, then $m=m_{1}m_{2}$ is right negative and hence $m$ is not dominant. Therefore $m_{2}\in \chi_{q}(1_{6}1_{12}\cdots1_{6k})m_{3}$.

Suppose that $m_{2}\in \mathscr{M}(L(m_{2}'))\cap \mathscr{M}(\chi_{q}(1_{6}1_{12}\cdots1_{6k-6})(\chi_{q}(1_{6k})-1_{6k})m_{4})$. By the Frenkel-Mukhin algorithm for $L(m_{2}')$ and Lemma $\ref{lemma}$, $m_{2}$ must have the factor $1_{6k+6}^{-1}$. But by the Frenkel-Mukhin algorithm and the fact that $m_{1}\in \chi_{q}(1_{0}1_{6}\cdots1_{6k-6})m_{3}$, $m_{1}$ does not have the factor $1_{6k+6}$. Therefore $m_{1}m_{2}$ is not dominant. Hence $m_{2}\in \chi_q(1_{0}1_{6}\cdots1_{6k-6})1_{6k}m_{4}$. It follows that $m_{2}=m_{2}'$.

By the Frenkel-Mukhin algorithm and the fact that $m_{1}\in \chi_{q}(1_{0}1_{6}\cdots1_{6k-6})m_{3}$, $m_{1}$ must be one of the following monomials,
\begin{align*}
& n_{1} =m_{1}'A_{1, 6k-3}^{-1}=1_{0}1_{6}\cdots1_{6k-12}1_{6k}^{-1}2_{6k-5}2_{6k-3}\cdots2_{6k+2l+5},\\
& n_{2} =m_{1}'A_{1, 6k-3}^{-1}A_{1, 6k-9}^{-1}=1_{0}1_{6}\cdots1_{6k-18}1_{6k-6}^{-1}1_{6k}^{-1}2_{6k-11}2_{6k-9}\cdots2_{6k+2l+5},\\
& \cdots\\
& n_{k} =m_{1}'A_{1, 6k-3}^{-1}A_{1, 6k-9}^{-1}\cdots A_{1, 3}^{-1}=1_{6}^{-1}\cdots1_{6k-6}^{-1}1_{6k}^{-1}2_{1}2_{3}\cdots2_{6k+2l+5}.
\end{align*}
It follows that the dominant monomials in $\chi_{q}(\mathcal T_{k, l+3}^{(s)})\chi_{q}(\mathcal T_{k, l}^{(s)})$ are
\begin{eqnarray*}
\begin{split}
&M=m_{1}'m_{2}', \ M_{1}=n_{1}m_{2}'=MA_{1, 6k-3}^{-1}, \ M_{2}=n_{2}m_{2}'=M\prod_{i=0}^{1}A_{1, 6k-6i-3}^{-1},\
\ldots, \\
&M_{k-1}=n_{k-1}m_{2}'=M\prod_{i=0}^{k-2}A_{1, 6k-6i-3}^{-1}, \ M_{k}=n_{k}m_{2}'=M\prod_{i=0}^{k-1}A_{1, 6k-6i-3}^{-1}.
\end{split}
\end{eqnarray*}
\end{proof}

\subsection{Proof of Theorem $\ref{M system of type G2}$}
By Lemma $\ref{lemma1}$, the dominant monomials in the $q$-characters of the left hand side and of the right hand side of every equation in Theorem $\ref{M system of type G2}$ are the same. Therefore the theorem is true.

\section{Proof of Theorem $\ref{irreducible}$}  \label{proof irreducible}

By Lemma $\ref{lemma1}$, the modules in the second summand of every equation in Theorem $\ref{M system of type G2}$ are special and hence they are irreducible. We only need to show that the modules in the first summand in every equation in Theorem $\ref{M system of type G2}$ are irreducible. Let $\mathcal{S}$ be a module in the first summand in an equation in Theorem $\ref{M system of type G2}$. It suffices to prove that for each non-highest dominant monomial $M$ in $\mathcal{S}$, we have $\chi_q(L(M))\not \subseteq \chi_q(\mathcal{S})$, see \cite{Her06}, \cite{MY12a}.

\begin{lemma}
We consider the same cases as in Lemma $\ref{lemma1}$. In each case $M_{i}$ are the dominant monomials described by that Lemma $\ref{lemma1}$.

\begin{enumerate}[(1)]
\item For $k\in \mathbb{Z}_{k\geq 1} \ l\in \{1, 2, 3\}$, let
\begin{equation*}
\begin{split}
&n_{r}=M_{r}A_{1, aq^{-2l-6r+4}}^{-1}, \quad r=1, 2, \ldots, k-1.
\end{split}
\end{equation*}
Then for $i=1, 2, \cdots, k-1$, $n_{i}\in \chi_{q}(M_{i})$ and $n_{i}\not \in \chi_{q}(\mathcal T_{k, l}^{(s)})\chi_{q}(\mathcal T_{k, 0}^{(s+6)})$.

\item For $k, l\in \mathbb{Z}_{\geq 1}$, let
\begin{equation*}
\begin{split}
&n_{r}=M_{r}A_{1, aq^{s+6k-6r+3}}^{-1}, \quad r=1, 2, \ldots, k-1.
\end{split}
\end{equation*}
Then for $i=1, 2, \cdots, k-1$, $n_{i}\in \chi_{q}(M_{i})$ and $n_{i}\not \in \chi_{q}(\mathcal T_{k, l+3}^{(s)})\chi_{q}(\mathcal T_{k, l}^{(s+6)})$.
\end{enumerate}
\end{lemma}

\begin{proof}
We give a proof of the case of $\chi_{q}(\mathcal T_{k, l+3}^{(s)})\chi_{q}(\mathcal T_{k, l}^{(s+6)})$. The other cases are similar. By definition, we have
\begin{equation*}
\begin{split}
&T_{k,l+3}^{(s)}=1_{s}1_{s+6}\cdots1_{s+6k-6}2_{s+6k+1}2_{s+6k+3}\cdots2_{s+6k+2l+5},\\
&T_{k,l}^{(s+6)}=1_{s+6}1_{s+12}\cdots1_{s+6k-6}1_{s+6k}2_{s+6k+7}2_{s+6k+9}\cdots2_{s+6k+2l+5}, \\
&M_{1}=T_{k,l+3}^{(s)}T_{k,l}^{s+6}A_{1, aq^{s+6k-3}}^{-1}=T_{k,l+3}^{(s)}T_{k,l}^{s+6}1_{s+6k-6}^{-1}1_{s+6k}2_{s+6k-5}2_{s+6k-3}2_{s+6k-1}.
\end{split}
\end{equation*}
By $U_{q^{3}}(\widehat {\mathfrak sl}_{2})$ argument, it is clear that $n_{1}=M_{1}A_{1, aq^{s+6k-3}}^{-1}$ is in $\chi_{q}(M_{1})$.

If $n_{1}$ is in $\chi_{q}(T_{k, l+3}^{(s)})\chi_{q}(T_{k, l}^{(s+6)})$, then $T_{k, l}^{(s+6)}A_{1, aq^{s+6k-3}}^{-1}$ is in $\chi_{q}(T_{k, l}^{(s+6)})$ which is impossible by the Frenkel-Mukhin algorithm for $T_{k, l}^{(s+6)}$. Similarly, $n_{i}\in \chi_{q}(M_{i}), i=1, 2, \cdots, k-1$, but $n_{2}, n_{3}, \cdots, n_{k-1}$ are not in $\chi_{q}(T_{k, l+3}^{(s)})\chi_{q}(T_{k, l}^{(s+6)})$.
\end{proof}

\section{The dual $M$-system of type $G_2$} \label{dual M system}
In this section, we study the dual $M$-system of type $G_2$.

\begin{theorem}[{Theorem 7.2, \cite{LM13}}]
The module $\widetilde{\mathcal T}_{k, l}^{(s)}$, $s\in \mathbb{Z}, k, l\in \mathbb{Z}_{\geq 0}$ are anti-special.
\end{theorem}

\begin{lemma}[{Lemma 7.3, \cite{LM13}}]\label{lemma2}
Let $\iota: \mathbb{Z}\mathcal{P}\rightarrow \mathbb{Z}\mathcal{P}$ be a homomorphism of rings such that $Y_{1, aq^{s}}\mapsto Y_{1, aq^{12-s}}^{-1}$, $Y_{2, aq^{s}}\mapsto Y_{2, aq^{12-s}}^{-1}$ for all $a\in \mathbb{C}^{\times}, s\in \mathbb{Z}$. Then
$$\chi_{q}(\widetilde{\mathcal T}_{k, l}^{(s)})=\iota(\chi_{q}(\mathcal T_{k, l}^{(s)})).$$
\end{lemma}

\begin{theorem}\label{dual-M-system}
For $s\in \mathbb{Z}$, $k, l \in \mathbb{Z}_{\geq 0}$, we have the following system of equations which we call the dual M-system of type $G_2$.
\begin{eqnarray*}
[\widetilde{\mathcal T}_{k, l}^{(s)}][\widetilde{\mathcal T}_{k, 0}^{(s+6)}]&=&[\widetilde{\mathcal T}_{k+1, 0}^{(s)}][\widetilde{\mathcal T}_{k-1, l}^{(s+6)}]+[\widetilde{\mathcal T}_{0, 3k+l}^{(s)}] \quad (k \in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\}), \\
{[\widetilde{\mathcal T}_{k, l+3}^{(s)}]}[\widetilde{\mathcal T}_{k, l}^{(s+6)}]&=&[\widetilde{\mathcal T}_{k+1, l}^{(s)}][\widetilde{\mathcal T}_{k-1, l+3}^{(s+6)}]+[\widetilde{\mathcal T}_{0, l}^{(s+6k+6)}][\widetilde{\mathcal T}_{0, 3k+l+3}^{(s)}] \quad (k, l\in \mathbb{Z}_{\geq 1}).
\end{eqnarray*}
Moreover, every module in the summands on the right hand side of every equation in the dual M-system is irreducible.
\end{theorem}
\begin{proof}
The lowest weight monomial of $ \chi_q(\mathcal{T}_{k,l}^{(s)}) $ is obtained from the highest weight monomial of $ \chi_q(\mathcal{T}_{k,l}^{(s)}) $ by the substitutions: $1_s \mapsto 1^{-1}_{12+s}$, $2_s \mapsto 2^{-1}_{12+s}$. After we apply $\iota$ to $\chi_{q}(\mathcal T_{k, l}^{(s)})$, the lowest weight monomial of $ \chi_q(\mathcal{T}_{k,l}^{(s)}) $ becomes the highest weight monomial of $\iota(\chi_{q}(\mathcal T_{k, l}^{(s)}))$. Therefore the highest weight monomial of $\iota(\chi_{q}(\mathcal T_{k, l}^{(s)}))$ is obtained from the lowest weight monomial of $ \chi_q(\mathcal{T}_{k,l}^{(s)}) $ by the substitutions: $1_s \mapsto 1^{-1}_{12-s}$, $2_s \mapsto 2^{-1}_{12-s}$. It follows that the highest weight monomial of $\iota(\chi_{q}(\mathcal T_{k, l}^{(s)}))$ is obtained from the highest weight monomial of $ \chi_q(\mathcal{T}_{k,l}^{(s)}) $ by the substitutions: $1_s \mapsto 1_{-s}$, $2_s \mapsto 2_{-s}$. Therefore the dual M-system is obtained applying $\iota$ to both sides of every equation of the M-system.

The irreducibility of every module in the summands on the right hand side of every equation in the dual M-system follows from Theorem $\ref{irreducible}$ and Lemma $\ref{lemma2}$.
\end{proof}

\begin{example}
The following are some equations in the dual $M$-system of type $G_2$.
\begin{eqnarray*}
\begin{split}
&[2_{0}1_{7}][1_{1}]=[1_{1}1_{7}][2_{0}]+[2_{0}2_{2}2_{4}2_{6}],\\
&{[2_{0}2_{2}1_{9}]}[1_{3}]=[1_{3}1_{9}][2_{0}2_{2}]+[2_{0}2_{2}2_{4}2_{6}2_{8}],\\
&{[2_{0}2_{2}2_{4}1_{11}]}[1_{5}]=[1_{5}1_{11}][2_{0}2_{2}2_{4}]+[2_{0}2_{2}2_{4}2_{6}2_{8}2_{10}],\\
&{[2_{0}2_{2}2_{4}2_{6}1_{13}]}[2_{0}1_{7}]=[2_{0}1_{7}1_{13}][2_{0}2_{2}2_{4}2_{6}]+[2_{0}][2_{0}2_{2}\cdots2_{10}2_{12}],\\
&{[2_{0}2_{2}\cdots2_{20}1_{27}1_{33}]}[2_{0}2_{2}\cdots2_{14}1_{21}1_{27}]\\
&=[2_{0}2_{2}\cdots2_{14}1_{21}1_{27}1_{33}][2_{0}2_{2}\cdots2_{20}1_{27}]+[2_{0}2_{2}\cdots2_{12}2_{14}][2_{0}2_{2}\cdots2_{30}2_{32}].
\end{split}
\end{eqnarray*}
\end{example}

\subsection{The dual $m$-system of type $G_2$}
For $k, l \in \mathbb{Z}_{\geq 0}$, let $\widetilde{m}{_{k,l}} = \res(\widetilde{\mathcal{T}}{_{k,l}^{(0)}})$ be the restriction of $\widetilde{\mathcal{T}}{_{k,l}^{(0)}}$ to $U_q \mathfrak{g}$. By Theorem \ref{dual-M-system}, we have the following theorem.
\begin{theorem}\label{m system}
We have
\begin{eqnarray}
\chi(\widetilde{m}{_{k, l}}) \chi( \widetilde{m}{_{k, 0}}) &=& \chi(\widetilde{m}{_{k+1, 0}}) \chi( \widetilde{m}{_{k-1, l}}) + \chi( \widetilde{m}{_{0, 3k+l}}) \quad (k \in \mathbb{Z}_{\geq 1}, l \in \{1, 2, 3\}), \\
\chi( \widetilde{m}{_{k, l+3} }) \chi(\widetilde{ m}{_{k, l}} ) &=&  \chi( \widetilde{m}{_{k+1, l}} ) \chi( \widetilde{m}{_{k-1, l+3}} ) + \chi( \widetilde{m}{_{0, l}} ) \chi( \widetilde{m}{_{0, 3k+l+3}} ) \quad (k, l \in \mathbb{Z}_{\geq 1}).
\end{eqnarray}
\end{theorem}
We call the above system of equations the dual $m$-system of type $G_2$.

\section*{Acknowledgements}
Jian-Rong Li is very grateful to Professors David Hernandez, Bernard Leclerc, and Evgeny Mukhin for stimulating discussions. J.-R. Li is supported by ERC AdG Grant 247049, the PBC Fellowship Program of Israel for Outstanding Post-Doctoral Researchers from China and India, the National Natural Science Foundation of China (no. 11371177, 11401275), and the Fundamental Research Funds for the Central Universities of China (no. lzujbky-2015-78).

redpajama_set_name: RedPajamaArXiv


Dominic Chu is a senior markets correspondent for CNBC, located at the network's Global Headquarters in Englewood Cliffs, N.J. He appears during CNBC's Business Day programming and contributes to CNBC.com.
Previously, Chu was a New York-based markets correspondent for Bloomberg Television, where he covered the stock, bond, currency and commodities markets. During that time, he interviewed some of the world's top money managers and business executives, and he also was part of the team that covered Hurricane Sandy and the Boston Marathon bombings. In addition, Chu handled sports business reporting for the network.
Chu brings extensive knowledge of the financial markets, having worked in sales and trading for UBS Investment Bank, mutual fund management for Hennessy Advisors and investment management for Seascape Capital.
He has spoken at numerous industry conferences and was a regular contributor to radio and television outlets across the country.
Chu holds a Bachelor of Science degree in hotel administration from Cornell University.
Follow Dominic Chu on Twitter @TheDomino.
Privately held Purdue Pharma has agreed to pay $270 million to resolve claims that the company did help to fuel the opioid crisis. This is the first of over 1600 opioid-related lawsuits the company faces. CNBC's Dom Chu reports.
Ross Gerber, Gerber Kawasaki CEO and co-founder, and CNBC's Dom Chu sit down with CNBC's Kelly Evans to discuss how the market is responding to a possible U.S.-China trade deal and other risks.
CNBC's Dom Chu reports on sources saying that Boeing will update the Boeing 737 Max 8's software in 10 days. The planes have been grounded in a number of countries after two deadly crashes.
The S&P 500 and Nasdaq rose yesterday for a second straight session while the Dow was dragged down by Boeing following the fatal crash on Ethiopian Airlines. Investors are also monitoring the on-going Brexit votes in the UK as the exit deadline approaches. CNBC's Dominic Chu reports.
Shares of Altria are falling after National Cancer Institute Director Ned Sharpless was named as the acting FDA commissioner. CNBC's Dom Chu reports.
CNBC's Dominic Chu reports on the financial sector's struggles through the beginning of March.


Fact: Salt and pepper are the holy grail of seasonings. Fun fact: "kale, cauliflower, red pepper" is a tongue twister. Random fact: Chickpea is just another word for garbanzo bean. Now, fact check these snacks and read our label.
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Organic kale, organic chickpeas, organic sunflower seeds, organic bell peppers, organic cauliflower, organic almonds, nutritional yeast, organic lemon juice concentrate, organic potato, sea salt, organic black pepper, organic garlic powder, organic cayenne.


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text:
Office Market News
The commercial real estate sector continues to help hospitals across Poland in the fight against the pandemic The #property4heroes campaign in the autumn
Commercial real estate companies have once again been coming to the aid of hospitals across Poland. Dinners, healthy snacks, pizzas, sandwiches and plates of sushi are being sent to the doctors, nurses, midwives and paramedics working to save lives and safeguard the health of their patients, and who have no time to prepare their own lunch or dinner while on a 24-hour shift. During the current campaign 10,500 meals have already been supplied to hospitals.
All the meals supplied to the medical workers are purchased from the bars and restaurants in the office and retail buildings owned by the companies taking part in the campaign. This means that #property4heroes has an additional beneficial effect – by purchasing meals to support medical staff in their work, these companies are also supporting restaurateurs whose businesses have been badly affected by the Covid-19 restrictions.
CA Immo, Grupa Capital Park, Echo Investment, EPP, the Globalworth foundation, Griffin Real Estate, HB Reavis and Panattoni have all joined the #property4heroes campaign.
In the last ten months our lives have been dominated by the coronavirus pandemic, when we have had to constantly deal with difficult situations. It's easy in such circumstances to become indifferent – and to forget how important it is to help those engaged in vital work. Commercial real estate companies have continued their campaign to support medical personnel working on the frontline during the second wave of the virus. It is impossible to be left unmoved by the level of dedication and self-sacrifice these workers have shown. And this is a sentiment that is obviously shared by everyone in our sector, as it continues to lend its support to our heroic hospital workers.
Meals are being delivered to hospitals in Warsaw, Kraków, Wrocław, Kielce and Katowice. Dinners are also being provided to ambulance crews on standby at accident & emergency wards and at blood donation and transfusion centres. And 'office doctors' – the medical staff and doctors who look after the health of Warsaw's office workers – take food orders from medical staff so that meals can be sent out to different places every day as part of the campaign.
Working under a strict sanitary regime is a serious challenge for medical personnel. Patients infected with Covid-19 are being treated not only in hospitals and specialist wards, but in every medical centre across Poland. Doctors and nurses are falling ill all the time, which means that healthy medics have to be on duty for longer and have often ended up working in smaller teams. These are the people who treat patients who have suffered heart attacks, strokes and other serious conditions. They deliver babies and provide treatment for the newborns. They treat patients in psychiatric wards.
As the health workers who receive these meals point out, their workload has certainly not been reduced by the outbreak of the coronavirus. In each hospital and medical centre there are staff who are often required to work 24-hour shifts. The support of the real estate sector not only gives doctors, paramedics, nurses and midwives something to look forward to during the day, but it can also help them to concentrate on the most important aspect of their work – the health and lives of their patients.
The #property4heroes campaign is not just about providing meals. In the autumn 1,000 protective masks were donated to hospitals by the campaign, as well as protective clothing, sterilising lamps and coffee machines.
A record year for the largest Polish developer
Dimitris Raptis is appointed Chief Executive Officer of Globalworth Group.
The Hi Łódź! mixed-use development in the very heart of Łódź has been granted an occupancy permit
Globalworth takes on the management of Warsaw Trade Tower.
DL PIANO WITH A NEW TENANT
MORE OFFICE MARKET NEWS »
The real estate market's largest and most complex professional office market portal in Poland, which provides a comprehensive orientation to the office market participants, whether they are landlords, real estate agents, developers, facility management companies, media interested in the office market or individuals. The "green site" of our portal presents and publishes the office market events through the collection of the news, happenings and transactions. The details of the office buildings in our data base appear in tables and charts as well and thanks to this function our visitors have easy access to information such as: specific characteristics of a chosen city's office market, total office area, available office space, average rental cost per district or per area. The "useful information" section provides further information about the terms and definitions of the office market, what kind of formulas and calculation methods are used by the landlords and real estate brokers to define common areas, how BOMA calculates and what is the methodology of GIF and RICS. The knowledge or lack of knowledge of these terms may result substantial loss of money for both the landlords and for the potential tenants.

redpajama_set_name: RedPajamaCommonCrawl


Washington, D.C. Mayor Muriel Bowser (D) plans to distribute 76,000 kits of naloxone, a drug used to counter opioid overdoses, to police officers and community health organizations by Sept. 30.
The purchase of 76,000 naloxone kits marks an increase in the city's efforts toward overdose death prevention, as only 2,396 kits were distributed over the last nine months of 2017, where overdose deaths were at their peak, according to The Washington Post. In 2018, D.C. saw more deaths from opioid-related overdoses than homicides.
The distribution plan is a part of Bowser's "LIVE. LONG. DC" initiative, which was released in December and aims to cut opioid related deaths in D.C. in half by 2020. An updated version of LIVE. LONG. DC, released March 28, includes more detailed procedures for D.C. institutions to begin carrying out the initiative.
FILE PHOTO: KIRK ZIESER/THE HOYA | The "LIVE. LONG. DC" initiative, which aims to cut opioid related deaths in Washington, D.C., in half by 2020, includes detailed procedures for the distribution of naloxone kits.
The update was released after complaints made by D.C. officials stated that the original document was too vague and lacked specific details such as timelines, according to The Washington Post.
In addition to the distribution of naloxone, the update includes an April 30 deadline for the start of another program which allows recent overdose victims to be treated with buprenorphine, a medication that diminishes opioid cravings as part of long-term addiction treatment. MedStar Washington Hospital Center, United Medical Center and Howard University Hospital will implement the initiative first.
To receive input on the plan, the D.C. Council Judiciary and Public Safety and Health committees held a joint public oversight roundtable Jan. 28, where they received feedback from the public, the mayor's office, the Department of Behavioral Health and the Metropolitan Police Department. The roundtable lasted for more than nine hours and covered topics such as street outreach to heroin users and the possibility of establishing government-supervised areas where residents can inject heroin, according to The Washington Post.
Opioid use and overdose prevention policies must treat addiction as a health issue, according to Regina LaBelle, program director at the O'Neill Institute for National and Global Health Law. The O'Neill Institute, which is housed at the Georgetown University Law Center, seeks to develop solutions to national and international health concerns such as the opioid epidemic.
LaBelle, who is also the former chief of staff of the White House Office of National Drug Control Policy, responded to potential opposition to the initiative in a testimony presented at the roundtable. The number of opioid-involved overdose deaths in the District more than tripled from 2014 to 2017. The uptick in overdoses occurred despite emergency medical responders providing residents with prompt services, according to LaBelle.
"It continues to be important, therefore, to get naloxone into the hands of as many people as possible, including law enforcement and community groups such as syringe exchange programs," LaBelle said during the testimony.
The Metropolitan Police Department and Bowser's office initially perceived the initiative as expensive. However, costs of naloxone can vary and manufacturers of the product have already put in place programs to make the drug more accessible for purchase by the government, according to US News.
The cost of naloxone varies from $12 generic injectable doses to a $150 nasal spray containing two doses, according to LaBelle. Manufacturers of the most expensive naloxone product, the auto-injector, announced that by mid-2019, their product would be available for purchase to government agencies for $178, significantly less than the usual retail cost of $4,100.50, according to LaBelle.
Other objections to the widespread distribution of naloxone among law enforcement and community healthcare centers included potential criminal liabilities. However, good samaritan laws and laws designed to protect healthcare professionals have been established in many states to protect those who administer the drug, according to the United States Surgeon General Jerome Adams.
Citizens can also play a role in administering the drug to prevent overdose deaths, according to to Adams. There are two FDA-approved naloxone products for community use that are available by prescription.
Too few community members are aware of the drug's accessibility, Adams wrote in a 2018 Naloxone Advisory Statement.
Research suggests that an effective measure to decrease overdose deaths in a community is to provide naloxone and overdose education to its members, according to Adam's statement.
To further increase the drug's accessibility, the FDA is currently developing an over-the-counter version of the drug, available for purchase by the public. People who are or who know someone at risk for opioid overdose can already go to a pharmacy or community-based program to receive naloxone and naloxone administration training without a patient-specific prescription, according to Adams.


text:
Browsing Lambda Alpha Journal of Man 3, no.1, 1971 by Subject
Lambda Alpha Journal
Lambda Alpha Journal of Man 3, no.1, 1971
Browsing Lambda Alpha Journal of Man 3, no.1, 1971 by Subject "Social anthropology"
Claude Levi-Strauss: Anthropologist as everyman
Von Sturmer, J.R.; Bell, J.H. (Lambda Alpha Anthropology Honors Society at Wichita State University, 1971)
Lambda Alpha Journal of Man, v.3, no.1 (complete version)
Lambda Alpha National Collegiate Honors Society for Anthropology (Lambda Alpha Anthropology Honors Society at Wichita State University, 1971)
This issue includes a guest paper on Claude Levi-Strauss by J.R. von Sturmer and J.H. Bell (Australia) and the listing of Master's thesis titles awarded by American Universities.


text:
Home - Vocations - Priests
Seminarians must not be 'St John Boscos by day but John Travoltas by night'
Madelaine Teahan - Catholic Herald - Mon, Aug 26th 2013
The vocations director of Southwark Diocese has established a Vocations Missions Team for would-be seminarians in an effort to support their discernment and prevent them from being "John Boscos by day and John Travoltas by night".
Fr Stephen Langridge has invited eight men who are considering entering the seminary to participate in nine months of missionary work that will include evangelising people on the streets.
He said an integral part of the programme was to emphasise the need for Catholics to grow in virtue.
He said the sacraments could not "magically" transform a person on their own and growth in virtue had to come also from within, by young men choosing to embrace the good and rejecting the bad.
Fr Langridge added that men considering the seminary could not be "John Bosco by day but John Travolta by night" and therefore must resist temptations such as excessive drinking.
He said that beginning in September this year, the men would receive "intense spiritual formation so that they are rooted in the call to holiness and a personal relationship with Christ". Among the members of the team are a 19-year-old from Kerala, a young man who has worked as a missionary in the Congo and another who has served as a soldier in Afghanistan.
On his blog Fr Langridge wrote: "I would ask your prayers please for a new project here at the Vocations Centre. Next year we will have eight young men living here forming our Vocations Mission Team. They will receive formation in the faith, in the spiritual life and in community living."
He continued: "What will set them apart from other similar projects, however, is that they will also go out to evangelise in schools, parishes, universities and on the streets."
Fr Langridge said he hoped that young people would take particular notice of other young men who are reaching out to evangelise them.
Fr Langridge said: "Pope Francis says it is not enough to open the door in welcome, we have to go out through that door in order to take Christ to others. The plan is for the experience of a year with our mission team to ensure that evangelisation becomes part of the DNA of our future priests."
The boys will live and work together in Southwark Vocations Centre in Whitstable, Kent, for the full nine months.


Cheat Summoners War will allow you to get free resources and accelerate the passage of the game. To improve your island and summoned creatures, you will need a lot of crystals and runes Summoners War hack, which you can now get for free in the game. To do this, you do not need to download the mods, or get root or jailbreak device rights. It is enough to study all the chapters of the article and learn how to enter secret codes to simulate gaming purchases, this will save a lot of rubles.
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This will allow you to significantly advance in the game and raise your level. Each character has its own element and class, so for victories you will not just have to have strong characters, you need to be able to use them correctly. Get experience in the game, having enough runes, speed of passage and training will give you even more fun with the game.


Ravens 49ers Super Bowl: The small decisions that cost San Francisco the big game.
The small decisions that cost the 49ers the Super Bowl.
San Francisco 49ers head coach Jim Harbaugh reacts to a play during Super Bowl XLVII against the Baltimore Ravens on Sunday.
Super Bowl XLVII turned out to be a close game that hinged on a few critical plays at the end—shocking, given the score when the power went out. But it also hinged on decisions that didn't seem very significant at the time but ended up having an enormous effect on the outcome.
One of the more obvious foul-ups was San Francisco's mismanagement of its timeouts in the second half. If the 49ers had any hope of coming back from down 22, it was certainly going to be close at the end, and timeouts were going to be invaluable. Colin Kaepernick, though, burned the 49ers' first timeout just before a first-and-10 play near midfield at the halfway point of the third quarter. It seemed as if he simply didn't like the play call or the match-ups because there was time left on the play clock. The second timeout was called by Jim Harbaugh to prevent a delay-of-game penalty on the 49ers' final drive that would have set up a third-and-goal from the 10 instead of from the 5. This was a debatable call. On the one hand, San Francisco would've been pushed 5 yards farther from the end zone. On the other, since Baltimore was taking away the run and forcing quick passes into man-to-man coverage, the extra space might have actually made things easier for the offense.
Jim Harbaugh made another mistake risking a timeout on the challenge of a first-down ruling halfway through the fourth quarter. He won the challenge, but that only set up third-and-inches, a situation that ends up being converted more than 80 percent of the time (and which the Ravens did ultimately convert). There wasn't enough upside to justify the risk that there wouldn't be enough evidence to overturn the ruling.
Had the 49ers retained all three timeouts, they would've gotten the ball back with about 1:22 to play and great field position, assuming they would've forced a Baltimore three-and-out. With two timeouts, San Francisco could have had 43 seconds left after a stop. In fact, with at least two timeouts, San Francisco would've had enough clock to kick the field goal on fourth down of what turned out to be its final possession, force a Baltimore punt, and then kick another field goal to win.
Here's how costly those timeouts were. First, consider the field goal/stop/field goal scenario. Typically, offenses needing a field goal to survive can score a three-pointer slightly more than 30 percent of the time when starting near their own 20-yard line. With about 45 seconds left to play, they can score about 15 percent of the time when starting near their own 20. If the 49ers played for the touchdown, they would have gained possession near midfield at worst. A touchdown from there with about 1:40 to play is about a 35 percent proposition, but with about 45 seconds to play, it's about 20 percent. When the 49ers called their first timeout and when Jim Harbaugh challenged that first-down ruling, it was impossible to know just how crucial those timeouts would become. These numbers, though, show just how valuable they can be, particularly when compared to the importance of a no-account first-down play in the third quarter.
The Ravens' intentional safety, by contrast, was probably a smart decision. At the time, I thought it was a mistake because it put San Francisco in a position where a field goal could tie the game. But I didn't realize how much time punter Sam Koch would be able to burn. The Ravens ran 8 seconds off the clock, thanks partially to a quirk in the rules. Since the penalty for holding in the end zone is, well, a safety, the Baltimore linemen could hold all night long without repercussion. Had a flag been thrown, the outcome would have been exactly the same.
The Ravens' fake field goal in the second quarter on fourth-and-9 at first made me scratch my head. The break-even chance of success to make the gamble worthwhile was 39 percent based on point expectancy and 62 percent based on win probability—the number is higher for win probability because it considers the fact that Baltimore was already ahead by 11 points. There aren't enough relevant cases to know what the typical chance of success would be here, but this is an example of why going for it on fourth down is far more lucrative than coaches think. The failed conversion attempt left San Francisco with the ball at its own 5. Baltimore was able to get the ball back in great field position, which ultimately led to a touchdown. Of course, that's not going to happen all the time, but it happens enough to matter. In the end, I appreciated the decision to try the fake because it showed that John Harbaugh was wise enough to expect the 49ers to come back at some point and that his team would need more points to stay ahead.
Jim Harbaugh missed at least one fourth-down opportunity. In the second quarter, he passed up a fourth-and-2 from the Baltimore 9, but the field-goal attempt was likely the right call. There was only enough time for three more plays in the half. San Francisco had two timeouts but needed a conversion first to be able to take advantage of them. The missed opportunity was a fourth-and-2 from the Baltimore 16 toward the end of the third quarter. David Akers had just missed his attempt on fourth-and-7, but a running-into-the-kicker penalty moved the 49ers into a situation in which they should have gone for it. Two-yard conversions are successful 59 percent of the time, and the break-even success rate was only 37 percent, as the 49ers were down eight at the time, needing a touchdown and two-point conversion to tie.
Jim Harbaugh deserves credit for leading his team to a dramatic near-comeback. Maybe if he'd used his timeouts more judiciously and his downs more aggressively, his team could've pulled off the biggest miracle win in Super Bowl history.


I am officially hooked on the Sin Brothers Series! I can't put these books down, and the suspense of the overall plot line just keeps building and building! In my opinion, the Sin Brothers Series is top notch when it comes to romantic suspense! It has all of the elements that I look for in an excellent story; well rounded characters, beautifully developed relationships, fascinating plot twists, some of the steamiest love scenes, and tender romantic entanglements!
In Blind Faith, we get a chance to know Nathan "Nate" Dean, the second oldest of the brothers. He comes across a video journal by the malevolent Dr. Madison that says Nate impregnated her daughter, Audrey five years ago. Audrey was Nate's first and only love. She broke his heart when she broke off the relationship with him, and he never thought that he'd see her again. However, with the possibility that Nate might have a child, he decides to travel to Washington D.C, where Audrey currently resides, so that he can uncover the truth.
Audrey is working closely with a powerful senator to take down the commander and the corrupt business that he is running. As she works closely to double cross the commander, she is surprised when Nate shows up at her apartment. She pleads with him to leave, because it is too dangerous for him to be around her. The commander has men tailing her. Audrey never stopped loving Nate and left him, in order to protect him. When Nate confides in Audrey about the kill chips, she decides that she wants to help him get the codes.
Imminent Danger lurks all around Nate and Audrey. Not only is the commander and his soldiers after them, but peers in Audrey's immediate professional circle begin turning up dead, and the killer is trying to frame her. Will Nate and Audrey be able to rekindle their romance, and if so, how long will they live to see another day?
Blind Faith was an excellent read!!! I loved both Nate and Audrey! It was such a thrill getting to read more about Matt, Shane, Lainey and Josie as well. The loyalty and sense of family in their group is heartwarming. "Never Alone," is the Dean Brothers motto, and adds another wonderful dimension to an already fabulous series. Ms. Zanetti gave us plenty of passionate love scenes, crazy plot twists, and an action packed addictive plot line! I give, Blind Faith, by Rebecca Zanetti, 4.5 Tempting and Teasing Passion Filled, Spine Tingling, and Red Hot Addictive Stars!
I was blown over with amazement when I read Nate's book. I loved the couple together and Audrey was a great heroine. Loved your quote pick!
Great review. The couple in this book sounds exciting to read about.
I really love your reviews Lindy Blind Faith sound like it would be my favorite book from the sin brothers series I hope I can get these out from the library to check them out because Nate and Audrey sound like the perfect couple. Thanks for the awesome giveaway!
I just started this audiobook today. So far, I'm loving it.
I love all her books and this series is a great one.
I love this series so much. It's so fantastic. I totally agree this series is the perfect combination of characters and suspense. Great review. I am dying for the upcoming 4th book!


Captain Hook's restaurant, located on the boardwalk at Rye Playland in Rye, NY is accepting EPKs from artists in all genres to perform Thursdays-Sundays this summer (May - September). On a good night the boardwalk can see upwards of 20,000 people. This is a fantastic promotional opportunity.
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Artists must supply ALL of their own sound/backline. This is a family stage so NO profanity of any kind.
Please note: There are to be NO phone calls to the agency in regard to this or any other submission.


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text:
Annual Tour de Parks Bike Ride Set for Sunday, March 22, as fundraiser for The Legacy Trail
February 27, 2020 February 27, 2020 by Staff Reports
Participants may choose among 21-mile, 35-mile and 62-mile options
Riders participate in the 2019 Tour de Parks. Photo courtesy Friends of the Legacy Trail
The annual Tour de Parks bike ride is set for Sunday, March 22, the Friends of The Legacy Trail has announced.
A fundraiser for The Legacy Trail and its connectors, the ride will start and finish at the historic Venice Train Depot, which is located at 303 E. Venice Avenue in Venice, a news release explains. Participants have the options of a 21-mile, 35-mile, or 62-mile ride along The Legacy Trail and to other parts of the county, the release adds.
"The ride goes through or passes 12 parks along the route and provides a beautiful, scenic experience for cyclists of all levels," the release points out.
The event is coordinated and staffed by Friends of The Legacy Trail, Sarasota Manatee Bicycle Club and the Sarasota County Parks, Recreation and Natural Resources Department, the release notes. Last year's event had more than 950 riders, the release adds.
"This immensely popular ride gives cyclists an opportunity to enjoy the outdoors, get to know the cycling routes in the area, and raise funds for an important cause," said Bud Gaunce, event director and past president of the Sarasota Manatee Bicycle Club, in the release.
This is the 10th year of the ride, the release notes.
"This year will be especially meaningful as construction begins on extension of the Trail to Payne Park in downtown
Sarasota and development of a connector to the City of North Port," said Louis Kosiba, president of Friends of the Legacy Trail, in the release. "Riders, runners, walkers and whole neighborhoods are eagerly waiting for the work to begin."
A rider celebrates completion of the 2019 Tour de Parks. Photo courtesy Friends of the Legacy Trail
The ride includes breakfast by First Watch and lunch by Mattison's, the release points out.
Doctors Hospital will provide drinks and snacks as the Rest Stop Sponsor, it adds.
More information about entry fees, registration, routes, and schedules may be found on the Tour de
Parks webpage at https://www.friendsofthelegacytrail.org/tour-de-parks-information/.
"Friends of The Legacy Trail, Inc. (FLT) is an all-volunteer 501(c)(3) nonprofit organization based in Sarasota.
FLT exists to support, promote, enhance, and protect The Legacy Trail, a Sarasota County Park, and its trail connectors now and for future generations," the release explains. "It has nearly 1,000 individual, family, and business members."
Categories Children and youth, City of North Port, City of Sarasota, City of Venice, Fundraisers, Neighborhoods, Nonprofit News, Parks and Recreation, Sarasota County, South County, Special Events, Sports, Tourism, Transportation Post navigation
County staff authorized to proceed with initial investigation into whether site of Central Energy Plant in downtown Sarasota could be used for 200-bed secured treatment center operated by Sheriff's Office
663,000-gallon spill of reclaimed water contained in stormwater retention pond, county staff reports


text:
NEWS CREATION SCIENCE UPDATE HUMAN STEWARDSHIP
Were Humans Created to Commune in Nature?
BY BRIAN THOMAS, PH.D. * |
In 2008, scientists found that taking a walk in a park setting improves concentration.1 This year, a study in the Journal of Environmental Psychology discovered that being outside among natural surroundings also provides people with an inner sense of vitality or energy.2 Why is there this connection between nature and human well-being?
Researcher Richard Ryan of the University of Rochester stated in a university press release, "Nature is fuel for the soul," which in turn positively affects a person's health. He continued, "Research has shown that people with a greater sense of vitality don't just have more energy for things they want to do, they are also more resilient to physical illnesses."3 Ryan also referred to earlier research showing that people are more caring and generous when they are surrounded by natural settings.
The Journal of Environmental Psychology study carefully distinguished between increased vitality that comes just from physical activity, and a separate sense of well-being. The results showed that just being in the presence--or even in the remembrance--of nature, without any exercise, has its own energizing effect.
If humans are nothing more than well-organized atoms, why do they have any awareness of an immaterial "soul," let alone a sense of it being invigorated by natural settings? Evolution provides no logical answer, but the creation model does.
When the first man was made, his primary task was given in Genesis 2:15: "The LORD God took the man, and put him in the garden of Eden to dress [work] it and to keep [care for] it." Many can understand the pleasures that have motivated domestic gardeners worldwide for millennia. Studies that show the varieties of benefits from being outside all make sense in the context that humans were designed to commune with God and carry out work in a natural setting.
Thomas, B. A Dose of Nature Improves Concentration. ICR News. Posted on icr.org October 21, 2008, accessed June 4, 2010.
Ryan, R. M. et al. 2010. Vitalizing effects of being outdoors and in nature. Journal of Environmental Psychology. 30 (2): 159-168.
Spending Time in Nature Makes People Feel More Alive, Study Shows. University of Rochester press release, June 3, 2010.
* Mr. Thomas is Science Writer at the Institute for Creation Research.
Article posted on June 10, 2010.
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Greg's Lawn & Landscaping is a Full-Service Landscaping and Snow Removal Company based in Cedar Rapids, Iowa.
ecological, economical and ethical landscaping, lawn-care, snow management and parking lot sweeping services to both commercial and residential customers in Iowa and its surrounding counties.
Protect your landscape with a horticulturally-sound maintenance program tailored to your specific needs and site conditions.
Our professional landscape designers and construction team will work with you to create the outdoor living space of your dreams.
We design, install, retrofit, repair and maintain all types of smart irrigation and sprinkler systems.
We service residential, commercial, industrial, retail, multi-family, campus, and municipal sites. Call us for your snow and ice management needs.
No matter the size or complexity of the job, we're committed to excellence and guarantee the greatest return on your landscape and snow management investment. Our core tenets are based on integrity, efficiency, quality, reliability, and value-added services. A family-owned business with the experience and equipment that rivals the national players, you get the best of both worlds at Greg's Lawn and Landscape.
Two of only five CSP's [Certified Snow Professional] in all of Iowa, we pride ourselves on professionalism, innovation and safety. Our 35 year track-record speaks for itself, but if you don't want to take our word for it, ask the thousands of satisfied customers we had the pleasure of serving for the last 35 years.
I have been working with Greg and Nick for a few months and I am very impressed with the quality, skills and especially the customer service they provided me. The crew came together very quickly to clean up the Pool Park at Pool Tech before one of their VIP Night's. It was done very professionally and on a timely matter. I will continue to refer them out to others who are in need of lawn care and landscaping. The plants they added in are absolutely beautiful.
Hi Zach: I wanted to "THANK YOU" and your "Landscape Crew, Fertilizing Crew & Lawn Mowing Crew" on behalf of STONE CREEK FIRST OWNERS ASSOCIATION. I want to "Compliment" your Fertilizer Crew as our Association did NOT have any dandelions this year. Your fellows did a "Great Job" of covering our lawn as I have not received one complaint regarding weeds or dandelions, so "THANK YOU" for this Great Service. Austin & his Lawn Mowing Crew do a great job week after week & again our Association appreciates it. The "Planters" the fellows just finished look awesome and they did a great job of cleaning up after they finished. Thanks to Greg's Lawn fellows one and all, yourself included for keeping our area looking GREAT.
I've been analyzing and tracking both landscapers and snow pros for over 25 years and Greg's Lawn and Landscaping is one of the best in the Country! He's also one of a handful of professionals to have earned his CSP [Certified Snow Professional] credential. He is one of the most customer-centric companies in the entire industry.
An article from Snow Business Magazine, a SIMA publication. December 2018.
Choosing the right Snow Removal & Ice Management Company.
Core Aeration and Overseeding: What, When, and Why?


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Tua Tagovailoa Expected to Miss Remainder of the Year
Tua Tagovailoa is expected to miss the rest of the college football season after suffering a dislocated right hip late in the first half against Mississippi State. Tagovailoa was carted off the field with three minutes left in game in visible pain. After the game he was taken via helicopter to St. Vincent's Medical Center in Birmingham Alabama.
Tagovailoa finished the game completing 14/18 passes for 256 yards and two touchdowns. Prior to the injury Tagovailoa was considered a lock to be the number one quarterback taken in the 2020 NFL Draft. Should he forgo his final season and declare for the draft, Tagovailoa will finish first in passing touchdowns at Alabama with 85 and third in passing yards with 7,186. Alabama team surgeon Dr. Lyle Cain released a statement on the status of Tagovailoa.
Filed Under: Alabama Crimson Tide, Tua Tagovailoa